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Both alternatives can be proved using the same smallness mechanism. We first establish the initial-object lemma for complete categories with a weakly initial set, then obtain the two adjoint functor existence theorems and their applications.

Let $\mathcal{E}$ be a locally small category with all small categorical limits and a weakly initial set $(W_i{)}_{i\in I}$. Its product in a category $W={\prod }_i{W}_i$ is weakly initial: given any object, choose a map from some $W_i$ and compose with the corresponding projection. Take the simultaneous equalizer $e:E\to W$ of every endomorphism of $W$ and its identity; local smallness makes this a small categorical limit. Then $E$ is also weakly initial. If $a,b:E\to X$, form their equalizer $j:Y\to E$ and choose $t:W\to Y$. Since $ejt$ is an endomorphism of $W$, the construction of $e$ gives $ejte=e$. Cancelling the monomorphism $e$ yields $jte=1_E$. Thus $j$ is both a monomorphism and a split epimorphism, hence invertible, and $a=b$. There is therefore exactly one arrow from $E$ to every object: $E$ is initial.

The Freyd general adjoint functor theorem says that for a functor $U:\mathcal{A}\to \mathcal{B}$ between locally small categories, with $\mathcal{A}$ complete, a left adjoint exists exactly when $U$ preserves all small categorical limits and satisfies the solution-set condition. That condition requires, for each $B$, a set of arrows $u_i:B\to U{A}_i$ such that every $u:B\to UA$ factors as $U(f)u_i$ for some $i$ and $f:A_i\to A$.

For sufficiency, the comma category $(B\downarrow U)$ is locally small and complete: form a diagram's categorical limit in $\mathcal{A}$, and use categorical limit preservation to obtain its arrow from $B$. The solution set is a weakly initial set in this comma category. The lemma provides an initial object $(LB,{\eta }_B:B\to ULB)$. Its universal property givesFor $h:B\to B^{\prime }$, define $Lh$ by $U(Lh){\eta }_B={\eta }_{B^{\prime }}h$. Uniqueness proves the identity and composition laws and naturality of the displayed bijections, hence $L⊣U$. Conversely, an adjunction makes $U$ preserve categorical limits, since the corresponding hom-set functors turn a categorical limit cone into a categorical limit of sets. The single arrow ${\eta }_B$ is a solution set. This proves both directions of the theorem.

Now let $F:\mathcal{C}\to \mathcal{D}$ have small domain and codomain. The functor categories $[\mathcal{D},\mathbf{Set}]$ and $[\mathcal{C},\mathbf{Set}]$ are locally small, and their categorical limits are pointwise. Thus the precomposition functor $F^{\ast }$ preserves categorical limits and its domain is complete. To verify its solution-set condition, fix $X:\mathcal{C}\to \mathbf{Set}$ and an arrow $x:X\to F^{\ast }Y$. Let $Y_0\subseteq Y$ be the subfunctor generated by all elements $x_c(a)$, where $c\in \mathcal{C}$ and $a\in X(c)$. Explicitly,These sets are stable under every $Y(h)$. If $\kappa$ is an infinite cardinal bounding $|\mathrm{Mor}\mathcal{D}|$ and $\left|{\coprod }_{c}X(c)\right|$, every $Y_0(d)$ has cardinal at most $\kappa$. Replace each such set by a subset of a fixed set of size $\kappa$. There is only a set of choices of these subsets, of their action maps for the set of arrows of $\mathcal{D}$, and of maps from $X$ to their restrictions. These coded pairs form a solution set: $x$ factors through $F^{\ast }{Y}_0$ and then the inclusion $Y_0\hookrightarrow Y$. The general theorem now yieldsIts values may also be written ${\mathrm{Lan}}_{F}X(d)={\mathrm{colim}}_{(Fc\to d)}X(c)$, the usual left Kan extension formula.

For the other alternative, the categorical limit form of the Special adjoint functor theorem says that a limit-preserving functor $U:\mathcal{A}\to \mathcal{B}$ has a left adjoint if $\mathcal{A}$ is locally small, complete and well-powered, has a cogenerating set $(Q_i{)}_{i\in I}$, and $\mathcal{B}$ is locally small. A cogenerating set means that maps into its members distinguish any two parallel arrows. The dual theorem says that a colimit-preserving functor from a locally small, cocomplete, well-copowered category with a small generating family has a right adjoint. These conclusions are also necessary in their preservation clauses.

Here is a proof of the special theorem rather than an invocation of the general theorem without its required solution set. Given $(A,x:B\to UA)$, intersect all subobjects $m:M\hookrightarrow A$ through which $x$ factors after application of $U$. There are only a set of them by well-poweredness. Their intersection is a wide pullback, preserved by $U$, so gives a smallest supporting pair $(A_0,x_0)$. If $p,q:A_0\to Q_i$ satisfy $U(p)x_0=U(q)x_0$, their equalizer is a supporting subobject. Minimality makes that equalizer invertible, whence $p=q$. Consequentlyis injective. The cogenerating property makes the evaluation arrow from $A_0$ into the product in a category of the $Q_i$, one factor for each map $A_0\to Q_i$, a monic arrow. Its indexing set at each $i$ is identified with a subset of the fixed set $\mathcal{B}(B,U{Q}_i)$. Only a set of such products in a category can occur. Each has only a set of subobjects, and each possible subobject has only a set of arrows from $B$ into its image under $U$. These pairs give a weakly initial set in $(B\downarrow U)$, since every original pair receives the arrow from its minimal supporting pair. The complete-category lemma now gives the adjoint. Reversing all arrows proves the colimit form of the special theorem.

Apply that dual form to $F^{\ast }:[\mathcal{D},\mathbf{Set}]\to [\mathcal{C},\mathbf{Set}]$. Its domain is cocomplete and locally small, and $F^{\ast }$ preserves pointwise colimits. The set of covariant representable functors $\mathcal{D}(d,-)$ generates the domain: by the Yoneda lemma, maps from these representables probe every element at every object and hence distinguish distinct natural transformations. Finally, epimorphisms in this functor category are pointwise surjective. Indeed, their pointwise images form subfunctors; if an image is proper, the two inclusions into the pointwise pushout of two copies of the target over that image distinguish the original map, contradicting epicity. Quotients of a fixed functor are therefore specified by compatible equivalence relations on its sets of values, and form a set. Thus the domain is a well-copowered category. All special-theorem hypotheses hold, givingHere the Right Kan extension can be written ${\mathrm{Ran}}_{F}X(d)=\mathrm{Nat}(\mathcal{D}(d,F-),X)$. In particular both requested adjoints exist, although the two proofs use different smallness hypotheses.

Definitions organize category theory, but theorems give that organization mathematical force. I would therefore argue against treating the subject as an exception in which definitions are intrinsically more important than theorems. The distinction is especially misleading here because good categorical definitions are designed to expose hypotheses from which substantial theorems follow.

A category records objects, composable arrows, identities and associative composition. A functor preserves that structure. This language allows one to compare otherwise different mathematical settings, but it does not itself say which constructions exist. For example, the definition of a product in a category is not a recipe for forming a Cartesian product of underlying sets: it requires a universal factorization of pairs of arrows. If two products in a category $(P,p_1,p_2)$ and $(Q,q_1,q_2)$ exist, the universal properties give unique arrows $f:P\to Q$ and $g:Q\to P$ respecting projections. The composites $gf$ and $fg$ respect the same projections as the identities, so uniqueness forces them to be identities. Thus a definition yields a genuine theorem: products in a category are unique up to unique compatible isomorphism. The proof matters because it licenses replacing a construction by any other realization of the same property.

The Yoneda lemma is a more substantial example. For a covariant representable functor $\mathcal{C}(A,-)$ and $X:\mathcal{C}\to \mathbf{Set}$, a natural transformation $t:\mathcal{C}(A,-)\to X$ is determined by $a=t_A(1_A)$. Naturality forcesConversely this formula defines a natural transformation for every $a\in X(A)$. Hence $\mathrm{Nat}(\mathcal{C}(A,-),X)\cong X(A)$. The definition of naturality is indispensable, but the theorem identifies every possible transformation, proves representables detect arrows, and explains why universal objects can be studied through hom-sets. Merely knowing the definition does not supply this conclusion.

Similarly, an adjunction is defined by natural hom-set bijections. The construction of a free abelian group gives $\mathbf{Ab}({\mathbb{Z}}^{(S)},A)\cong \mathbf{Set}(S,UA)$, because a group homomorphism is determined uniquely by its values on the basis. This makes a universal extension problem transparent. But the Freyd general adjoint functor theorem goes further: it turns categorical limit preservation and a solution set into an actual adjoint. Its smallness hypotheses cannot be discarded as bookkeeping. For instance, the underlying-set functor from finite groups to finite sets has no left adjoint at a singleton. A putative universal finite group with universal element of order $m$ could not map that element to a generator of a finite cyclic group of order greater than $m$. The theorem's hypotheses separate existence from attractive notation.

The monoidal coherence theorem offers another example. A monoidal category has associativity and unit isomorphisms satisfying the pentagon identity for a monoidal category and triangle identity for a monoidal category. Coherence proves that every formal reassociation and unit insertion produces the same structural map. This is what justifies calculating without displaying all parentheses; it is not true merely because one has named an associator. Nor does it identify arbitrary additional maps, such as distinct braidings.

Finally, the opposite category construction shows how a definition can economize on proofs: reversing arrows takes products in a category to coproducts in a category, initial objects to terminal objects and monads to comonads. Yet this economy works because theorems and their proofs are invariant under the reversal. The subject is powerful precisely when definitions isolate reusable structure and theorems establish its consequences. Separating their importance obscures that partnership.

Theorems matter at least as much as definitions: they show when a categorical description captures actual mathematical structure. Definitions in category theory are exceptionally economical, but economy alone does not explain their applications.

An adjunction compresses a universal mapping property into natural hom-set bijections. The free-group adjunction, for example, records that assigning images to generators extends uniquely to a homomorphism. That is a useful definition, but proving that a free group exists and satisfies this property still requires mathematics. The definition organizes that proof rather than replaces it.

The Yoneda lemma offers a sharper example. A representable functor is easy to define; the theorem identifying natural transformations out of it with elements of another functor gives the definition its computational force. The representation of a set-valued functor as a colimit of representables is then a substantive consequence, and the theorem that filtered colimits commute with finite limits turns it into a criterion for finite-limit preservation. None of these conclusions follows merely from choosing attractive terminology.

Likewise a monad packages operations and their compatibility, while the Eilenberg-Moore category defines the corresponding algebras. The Beck monadicity theorem answers a different question: whether a given category is really recovered by that algebraic description. Applying it to compact Hausdorff spaces needs a topological argument about closed quotient relations. The definitions would not by themselves predict or justify the result.

Finally, abelian categories put kernels in a category, cokernels in a category and exactness into a setting where homological arguments can be reused. Some consequences, such as image factorization, become short because strong structural content has deliberately been included in the definition. Yet left derived functors, long exact sequences and the detection of projective objects supply information about obstructions to exactness which the terminology alone cannot provide.

Definitions and theorems therefore play complementary roles. A good definition states what should be invariant across examples; a theorem proves which constructions preserve it, which examples satisfy it, and what can be deduced from it. Category theory makes that partnership unusually visible, rather than making proofs less important.

Every covariant representable functor preserves categorical limits: arrows from its representing object into a limiting object are exactly compatible cones of arrows into the diagram. Apply the density formula proved in the root solution. For any finite diagram $D:J\to \mathcal{C}$,The middle isomorphism uses exactly the permitted theorem that filtered colimits commute with finite limits in sets; it applies because $E^{\mathrm{op}}$ is filtered. All maps are the natural comparison maps, so $F$ preserves finite limits, including the terminal object. This proves $\boxed{\text{(iii)}\Rightarrow \text{(i)}}$ and completes the equivalence.

The comma category $(A\downarrow F)$ has objects $(C,u)$ with $C\in \mathcal{C}$ and $u:A\to FC$ a set map. A morphism $(C,u)\to (D,v)$ is an arrow $h:C\to D$ such that $Fh\circ u=v$. Composition and identities are those of $\mathcal{C}$. For $A=1$, this is the category of elements $E$ of $F$: objects are $(C,x)$ with $x\in FC$ and arrows satisfy $Fh(x)=y$.

Associate to $(C,x)$ the covariant representable functor $H_C=\mathcal{C}(C,-)$. An arrow $h:(C,x)\to (D,y)$ of $E$ gives, by precomposition, $H_D\to H_C$. Thus this is a diagram $H:E^{\mathrm{op}}\to [\mathcal{C},\mathbf{Set}]$. There is a compatible natural map $H_C\to F$ sending $f:C\to B$ to $Ff(x)$.

Colimits in this functor category are pointwise. At an object $B$, the induced map from the colimit sends a class represented by $(C,x,f:C\to B)$ to $Ff(x)$. It is surjective because $z\in FB$ is represented by $(B,z,1_B)$. It is injective because every representative with value $z$ is identified with that same canonical representative: the arrow $f:(C,x)\to (B,z)$ in $E$ becomes an arrow of the indexing diagram sending $1_B$ to $f$. These bijections are natural in $B$, provingSmallness of $\mathcal{C}$ and set-valuedness of $F$ ensure that $E$ is small, so the displayed colimit is defined. This proves the introductory density request before the three equivalent conditions.

The Yoneda embedding is an exact representation of a category, but does not by itself reduce the subject to the study of arbitrary functor categories. There are both a size issue and a structure issue.

Let $\mathcal{A}$ be a locally small category and set $yA=\mathcal{A}(-,A)$. For a categorical presheaf $P:{\mathcal{A}}^{\mathrm{op}}\to \mathbf{Set}$, a natural transformation $\theta :yA\to P$ is determined by $x={\theta }_A(1_A)$. Indeed, naturality along $f:B\to A$ forcesConversely this formula defines a natural transformation for each $x\in P(A)$, since functoriality gives naturality along every $g:C\to B$. Taking $P=y{A}^{\prime }$ givesThus the Yoneda embedding is full and faithful, and $\mathcal{A}$ is equivalent to its full subcategory of representable presheaves. The indexing category in the usual version is ${\mathcal{A}}^{\mathrm{op}}$, not necessarily a small category chosen independently of $\mathcal{A}$.

Local smallness makes the values $\mathcal{A}(B,A)$ sets; it does not make the collection of objects a set. For a large indexing category, the collection of natural transformations between arbitrary set-valued functors can itself be a proper class. For example, take the discrete category on the class of ordinals and the constant two-element functor. Its endomorphisms include an independently chosen function $\{0,1\}\to \{0,1\}$ at each ordinal. A small functor category cannot encode this collection without changing universes. The Yoneda lemma remains meaningful when the source is representable because the displayed bijection identifies its natural transformations with a set. It does not make the entire large functor category locally small.

Even for a small category, the representable image is usually a very special part of its presheaf category. If $\mathcal{A}$ is the terminal category, its presheaf category is $\mathbf{Set}$, but its representable image consists only of a singleton. The presheaf category has all small categorical limits and colimits, whereas an arbitrary full subcategory need have almost none. The Yoneda embedding preserves existing limits because representable functors preserve limits. It generally does not preserve colimits. For example, the coproduct of two singleton sets in category of finite sets is a two-element set, but evaluating the proposed presheaf coproduct at a two-element set givesThe pointwise coproduct therefore leaves the representable image. Identifying which presheaves are representable, and which ambient constructions return to the original category, is precisely substantive category theory.

The representation is nonetheless powerful. The Yoneda lemma replaces a morphism by all its probes, converts a universal property into a representation of a functor, and makes the presheaf category a natural place to add formal colimits. A small presheaf category is the free free cocompletion under small colimits: every presheaf is a colimit of representables indexed by its category of elements. But this is a controlled enlargement of the category, not an identification of its original objects with all functors. The stronger conclusion in the proposed assertion therefore does not follow from the full-faithfulness statement.

A regular category has finite limits, coequalizers of kernel pairs, and regular epimorphisms stable under pullback. Equivalently, every morphism factors as a regular epimorphism followed by a monomorphism, and these factorizations are stable under pullback. A regular functor preserves finite limits and regular epimorphisms, hence also preserves these image factorizations.

For this question, the capital regular category convention is the one in which every well-supported object is a well-pointed object in a category. Here an object $A$ is well-supported if $A\to 1$ is a regular epimorphism. It is well-pointed if a subobject $m:B\hookrightarrow A$ through which every point $1\to A$ factors must be the whole object. This convention is documented in Definitions 1.1–1.3. It imposes no such point-detection requirement on objects whose support is a proper subobject of $1$. In particular, it should not be replaced here by the stronger assertion that $\mathcal{A}(1,-)$ reflects every isomorphism.

First prove two facts about a capital regular category $\mathcal{A}$. Every well-supported object $A$ has a point. Otherwise $A\times A$ is well-supported and has no points, so its diagonal $A\hookrightarrow A\times A$ contains all points and is invertible. That makes $A$ a subterminal object; the regular epimorphism $A\to 1$ is then also monic and hence invertible, contradicting the absence of a point. If $e:X\to Y$ is a regular epimorphism and $y:1\to Y$ is a point, its pullback in a category is a well-supported object over $1$, so it has a point lifting $y$. Thus the representable functorpreserves regular epimorphisms; it already preserves all existing limits. Hence $\Gamma$ is a regular functor. Second, if $\Gamma X$ is a singleton, its unique point $x:1\to X$ is a monomorphism, $X$ is well-supported, and $x$ contains all points of $X$. Well-pointedness makes $x$ invertible. Thus $\Gamma$ reflects terminal objects, even though it need not reflect arbitrary isomorphisms.

Now let $\mathcal{C}$ be a small category that is a regular category. For each object $B$, its slice category $\mathcal{C}/B$ is small and regular: finite limits, images and the stability of regular epimorphisms are computed using the corresponding constructions in $\mathcal{C}$. By the permitted capitalization result choose a conservative functor that is regular,with ${\mathcal{A}}_B$ a small capital regular category. The base-change functor $B^{\ast }:\mathcal{C}\to \mathcal{C}/B$, given by $X\mapsto (X\times B\to B)$, preserves finite limits and regular epimorphisms. So doesTaking these as the coordinates defines a regular functorThe exponent is a set of indices, so this is a power of the Category of sets, with finite limits and regular epimorphisms computed coordinatewise.

To prove reflection of isomorphisms, suppose $R(f)$ is invertible for $f:A\to B$. In $\mathcal{C}/B$, the object $(A,f)$ is the pullback of $B^{\ast }f:(A\times B\to B)\to (B\times B\to B)$ along the point given by the diagonal $B\to B\times B$. Applying ${\Gamma }_B{J}_B$ shows that ${\Gamma }_B{J}_B(A,f)$ is the fibre of the bijection $R_B(f)$ over this diagonal point, and is a singleton. The preceding lemma gives $J_B(A,f)\cong 1$ through its canonical arrow to $1$. Since $J_B$ reflects isomorphisms, $(A,f)\to (B,1_B)$ is invertible. Its underlying morphism is $f$. Therefore

In the small-category setting of this representation result, the elementary condition for replacing the power by one copy of $\mathbf{Set}$ is:This is almost total support for a regular category. “Strict initial” means initial and every arrow into it is invertible; the condition allows the case in which all objects are well-supported and no initial object exists.

For necessity, let $V:\mathcal{C}\to \mathbf{Set}$ be regular and conservative. If $VA\ne \varnothing$, factor $A\to 1$ through its support of an object in a regular category $S\hookrightarrow 1$. Preservation of images gives $VS=1$. Conservativity makes $S\to 1$ invertible, so $A$ is well-supported. If $VA=\varnothing$, then for every $X$ the projection $A\times X\to A$ is sent to the bijection $\varnothing \to \varnothing$, hence is invertible. Its inverse followed by the other projection supplies an arrow $A\to X$. For any two such arrows, their equalizer is sent to a bijection onto $VA$, hence is invertible, proving uniqueness. Thus $A$ is initial. If $X\to A$ exists, then $VX\to \varnothing$ exists, so $VX=\varnothing$ and that arrow is sent to a bijection. Conservativity makes it invertible. Hence $A$ is strict initial.

For sufficiency, choose the permitted conservative regular functor $J:\mathcal{C}\to \mathcal{A}$ into a small capital regular category, and put $V=\Gamma J$. This is regular. A well-supported object of $\mathcal{C}$ is sent to a well-supported object of $\mathcal{A}$, which has a point. A non-well-supported object is strict initial by the condition, and is consequently a proper subterminal object. The functor $J$ preserves subterminality and reflects the noninvertibility of its arrow to $1$; its image is a proper subterminal object and has no points. Thus $VA$ is empty exactly on the non-well-supported objects.

Suppose $Vf$ is a bijection for $f:A\to B$. If both sets are empty, $A$ and $B$ are strict initial objects and $f$ is invertible. Otherwise $A,B$ are well-supported. Factor $Jf$ as a regular epimorphism followed by a monomorphism $m:I\hookrightarrow JB$. Surjectivity of $Vf$ makes every point of $JB$ factor through $m$; capitality of $\mathcal{A}$ makes $m$ invertible. Hence $Jf$ is a regular epimorphism. Its kernel pair $P=JA{\times }_{JB}JA$ is well-supported, because $P\to JA$ and $JA\to 1$ are regular epimorphisms. Injectivity of $Vf$ implies that every point of $P$ has equal coordinates, and therefore factors through the diagonal $JA\hookrightarrow P$. Since $P$ is well-pointed, this diagonal is invertible. Thus $Jf$ is also monic, hence an isomorphism, and conservativity of $J$ makes $f$ invertible. This proves sufficiency with actual point and image arguments, rather than asserting that global sections of every capital regular category is conservative.

Suppose $G$ is a representable functor, with a natural isomorphism $\theta :\mathcal{C}(R,-)\to G$. Put $r={\theta }_R(1_R)$. For any $(A,a)$ in the comma category $(1\downarrow G)$, the representing bijection gives a unique $f:R\to A$ with ${\theta }_A(f)=a$. By naturality, ${\theta }_A(f)=G(f)(r)$, exactly the condition for $f:(R,r)\to (A,a)$. Hence $(R,r)$ is an initial object.

Conversely, if $(R,r)$ is an initial object, thenis a bijection for each $A$: existence and uniqueness are precisely initiality applied to every $(A,a)$. For $h:A\to B$, $G(h){\theta }_A(f)=G(hf)(r)={\theta }_B(hf)$, proving naturality. Thus $r$ is a universal element and $\theta$ is a representation of a functor. Being representable is equivalent to the singleton comma category having an initial object.

If $G$ is a representable functor, it preserves small categorical limits. Part (b) makes $(1\downarrow G)$ have an initial object, which gives a weakly initial set.

Conversely, assume limit preservation and a weakly initial set in $(1\downarrow G)$. The needed comma-limit fact is explicit: for a small diagram in a category $(A_j,a_j)$, form $L={\lim }_j{A}_j$ in $\mathcal{C}$. The elements $a_j\in G(A_j)$ form a compatible family, hence determine a unique $a\in G(L)$ under $G(L)\cong {\lim }_{j}G(A_j)$. The projections from $(L,a)$ are a categorical limit in the comma category. This includes the empty diagram, since $G$ carries the terminal object to a singleton. Each comma hom-set is a subset of a hom-set in $\mathcal{C}$, so the comma category is also locally small.

Part (c) now gives an initial object of $(1\downarrow G)$, and part (b) supplies a representation of a functor. ThereforeThe size requirement is a genuine solution-set condition; limit preservation alone does not produce it in an arbitrary complete category.

Use the covariant Yoneda embedding $Y(A)=\mathcal{C}(A,-)$. An arrow $f:B\to A$, viewed as an arrow $A\to B$ in the opposite category, induces precomposition $h\mapsto hf$. Identities and composition are preserved because composition in $\mathcal{C}$ is associative.

The covariant Yoneda lemma is the bijectionIts inverse sends $x\in F(A)$ to ${\alpha }_B(h)=F(h)(x)$. This is a natural transformation: for $k:B\to C$, functoriality gives $F(k){\alpha }_B(h)=F(kh)(x)={\alpha }_C(kh)$. Conversely, naturality of an arbitrary $\alpha$ at $h:A\to B$ gives ${\alpha }_B(h)=F(h){\alpha }_A(1_A)$. Evaluation at the identity morphism therefore makes the two constructions inverse. Their formulas also prove naturality in $F$ and, contravariantly, in $A$.

Taking $F=Y(B)$ identifies natural transformations $Y(A)\to Y(B)$ with $\mathcal{C}(B,A)$. Thus $Y$ is full and faithful, and reflects isomorphisms; two image objects are isomorphic exactly when the original objects are isomorphic. It also carries existing colimits in $\mathcal{C}$ to categorical limits of covariant representable functors: maps out of a colimit are exactly compatible families of maps out of its diagram. Equivalently, this embedding preserves existing categorical limits in ${\mathcal{C}}^{\mathrm{op}}$. The direction matters: the paper uses covariant representables, rather than the more usual contravariant Yoneda embedding.

A Cartesian theory has a many-sorted first-order signature, with finite arities, and axioms expressed as sequents between Cartesian formulas. Those formulas use atomic relations and equality, truth and finite logical conjunctions. They also admit existential quantification when the quantified witness is provably unique: to form $\exists y\varphi (x,y)$, require $\varphi (x,y)\wedge \varphi (x,y^{\prime })⊢y=y^{\prime }$ relative to the theory. One can quantify a finite tuple in the same way. Arbitrary existential quantifiers, disjunction, negation and universal quantifiers are not formula constructors in this fragment. In particular falsity is not included as an additional finite-limit constructor. The universal force of a sequent comes from its free-variable context. This is the fragment whose categorical semantics requires exactly finite limits.

The Cartesian syntactic category ${\mathcal{C}}_{\mathbb{T}}$ has objects $\{\vec{x}\mid \varphi \}$, identifying harmless renamings and provably equivalent formulas-in-context. An arrow from $\{\vec{x}\mid \varphi \}$ to $\{\vec{y}\mid \psi \}$ is represented by a provably functional Cartesian relation $\theta (\vec{x},\vec{y})$: it entails $\varphi \wedge \psi$, is total on $\varphi$, and has a unique output tuple. Provable equivalence identifies arrows. Equality supplies identity arrows; composition quantifies the uniquely determined intermediate tuple in a conjunction. Totality and uniqueness prove that the composite is again functional.

The empty truth context is terminal. Products concatenate disjoint contexts and conjoin their formulas. A pullback of two functional relations adds the condition that their outputs agree; the common output is unique, so the existential quantification used to express it is Cartesian. This gives all finite limits and makes the usual equality diagrams into equalizers.

An internal model $M$ in a finite-limit category $\mathcal{E}$ interprets each sort by an object, functions by arrows and relations by subobjects. Equality is a diagonal and conjunction is a pullback intersection. A uniquely witnessed relation projects monomorphically onto the context, so its existential interpretation needs no general image operation: it is that mono. Evaluating formulas and functional relations yields a finite-limit-preserving functor $F_M:{\mathcal{C}}_{\mathbb{T}}\to \mathcal{E}$.

Conversely, the canonical syntactic model uses the single-sort truth contexts, term graphs and atomic subobjects. Its interpretation of a formula is the formula-in-context object itself. Applying any finite-limit-preserving functor supplies a model and preserves its axioms. These constructions are inverse up to the natural interpretation isomorphisms; model homomorphisms correspond to natural transformations. Therefore

For completeness, let a sequent in context be represented by subobjects $i:A\hookrightarrow C$ and $j:B\hookrightarrow C$ for its antecedent and consequent. Derivability is exactly the existence of a factorization $i=jv$ in the syntactic category. Consider the covariant representable functor ${\mathcal{C}}_{\mathbb{T}}(A,-)$, which preserves all existing limits, and hence corresponds to a set-valued model. In that model the element $i\in {\mathcal{C}}_{\mathbb{T}}(A,C)$ belongs to the antecedent, because it is the image of $1_A$. If the sequent holds, it also belongs to the consequent, so $i=jv$ for some $v:A\to B$. That is precisely a derivation of the sequent. Thus validity in all set-valued models implies Cartesian derivability. More concretely, every underivable sequent is refuted by this representable model. No assumption that every sort is inhabited is needed: some hom-sets may be empty.

A functor $U:\mathcal{C}\to \mathbf{Set}$ is a representable functor if some object $R$ admits a natural isomorphism $\mathcal{C}(R,-)\cong U$. Here the representable is covariant.

For the identity functor on the Category of sets, choose the singleton $1=\{\ast \}$. The evaluation mapsare bijections with inverse $x\mapsto (\ast \mapsto x)$. For $u:X\to Y$, evaluation of $uf$ is $u(f(\ast ))$, proving naturality. Hence the identity functor on sets is represented by a singleton.

Let $U\cong \mathcal{C}(R,-)$ be a representable functor, and let $(L,p_j)$ be the categorical limit of a small diagram $D:J\to \mathcal{C}$. A morphism $R\to L$ is uniquely equivalent to a family $f_j:R\to D(j)$ satisfying $D(u)f_j=f_k$ for every $u:j\to k$.

Such compatible families are exactly the elements of the categorical limit of the set-valued diagram $\mathcal{C}(R,D(-))$. Consequently the canonical comparisonis a bijection. Transporting it through the representing natural isomorphism proves that $U$ preserves every small limit that exists in $\mathcal{C}$. This proves that covariant representables preserve limits. For an empty diagram this says that maps into a terminal object form a singleton.

A representable functor $yC$ is an indecomposable projective object. Given an epimorphism ${\coprod }_i{B}_i\twoheadrightarrow yC$, evaluate at $C$. Epimorphisms and coproducts in a presheaf category are pointwise, so $1_C\in yC(C)$ is the image of some element of a particular $B_i(C)$. By the Yoneda lemma, that element defines $s:yC\to B_i$, and its composite into $yC$ corresponds to $1_C$, hence is the identity. The selected component is split epic. More generally, evaluation sends any epimorphism to a surjection, so a map from $yC$ lifts through any epimorphism; this also proves its ordinary projectivity.

Conversely, every presheaf $A$ has the canonical epimorphismwhose component is the natural transformation named by $x$. It is pointwise surjective, since an element at $D$ is reached from its own summand $(D,x)$ at $1_D$. If $A$ is indecomposable projective, one component $r:yC\to A$ has a section $s:A\to yC$. The endomorphism $sr$ of $yC$ is idempotent and therefore corresponds to an idempotent morphism $e:C\to C$.

If idempotents split in $\mathcal{C}$, choose $C\stackrel{p}{\to }D\stackrel{i}{\to }C$ with $ip=e$, $pi=1_D$. Then $A\cong yD$: the mutually inverse maps are $y(p)s:A\to yD$ and $ry(i):yD\to A$. ThusWithout that hypothesis the argument still proves that every such object is a retract of a representable. The initial presheaf is not indecomposable projective, since its identity is the empty-coproduct epimorphism and has no component to select.

For every $C$, the preceding part identifies the composite $\mathcal{C}(C,{\lim }_J-)$ with the covariant representable functor $[J,\mathcal{C}]({\Delta }_{J}C,-)$. A covariant representable functor preserves every existing categorical limit, since a morphism into a categorical limit is the same as a compatible family of morphisms into the diagram objects. The limit functor preserves limits. The hom-set detection of categorical limits givesIn particular, it preserves all small categorical limits. They exist in $[J,\mathcal{C}]$: compute each one at every $j$ using completeness of $\mathcal{C}$, and use the uniqueness of the pointwise mediating morphisms to obtain its functor structure and universal natural transformations. Another expression of the same result is the adjunction ${\Delta }_J⊣{\lim }_J$.

By the canonical colimit presentation of a covariant set-valued functor established in the first part,Each vertex is a representable functor. Under condition $(iii)$ the indexing category of elements, with its arrows reversed, is a filtered category, so this is a filtered colimit in a category. Therefore $(iii)\Rightarrow (iv)$, with the requested diagram itself providing the filtered presentation.

First address the unnumbered preliminaries. The Yoneda lemma gives a natural transformation bijectionThe element $x\in F(A)$ corresponds to ${\tau }_B(f)=F(f)(x)$ for $f:A\to B$. These bijections are natural in $A$ and $F$; the contravariant form replaces $\mathcal{C}(A,-)$ by $\mathcal{C}(-,A)$.

Let $E=(1\downarrow F)$ be the category of elements. Its objects are $(A,x)$ with $x\in F(A)$; an arrow $u:(A,x)\to (B,y)$ is $u:A\to B$ with $F(u)(x)=y$. Since $\mathcal{C}$ is a small category, $E$ is small. Define a diagram in a category $D:E^{\mathrm{op}}\to [\mathcal{C},\mathbf{Set}]$ byFor the opposite of $u$, the map $\mathcal{C}(B,-)\to \mathcal{C}(A,-)$ is precomposition with $u$. The Yoneda lemma gives a cocone under a diagram $D\to F$ whose $(A,x)$ component sends $f:A\to B$ to $F(f)(x)$.

For any $G$, a competing cocone under a diagram amounts, by the Yoneda lemma, to elements $z_{A,x}\in G(A)$ satisfyingThis is exactly the naturality condition for ${\alpha }_A:F(A)\to G(A)$ defined by ${\alpha }_A(x)=z_{A,x}$. It gives a unique natural transformation $\alpha :F\to G$ factoring the cocone. Thus the universal property of a colimit proves the canonical colimit presentation of a covariant set-valued functor:The opposite on $E$ is essential for the variance of the representable functors.

We now prove the four conditions equivalent by the cycle $(i)\Rightarrow (ii)\Rightarrow (iii)\Rightarrow (iv)\Rightarrow (i)$. Suppose $F$ is a finite-limit-preserving set-valued functor. For a finite diagram in a category in the comma category $(S\downarrow F)$, write its objects as $(A_i,s_i:S\to F(A_i))$. Take $L={\lim }_i{A}_i$ in $\mathcal{C}$. Since $F$ preserves this finite limit, the compatible maps $s_i$ determine a unique $s:S\to F(L)$ whose composites with $F(L)\to F(A_i)$ are $s_i$. The pair $(L,s)$ has the required categorical limit property in $(S\downarrow F)$, by the same universal properties. For the empty diagram, $F(1)$ is a singleton, so there is exactly one map $S\to F(1)$; this provides the terminal object. Hence $(i)\Rightarrow (ii)$ for every set $S$.

Suppose $F\cong {\mathrm{colim}}_{j\in J}{R}_j$, where $J$ is a filtered category and every $R_j$ is a representable functor. Every covariant representable functor $\mathcal{C}(A,-)$ preserves all existing categorical limits: maps from $A$ into a limiting object are exactly compatible families of maps into its diagram.

For a finite diagram in a category $K:I\to \mathcal{C}$, use pointwise colimits in the functor category, preservation by each $R_j$, and the permitted result that filtered colimits commute with finite limits in sets to obtainThe isomorphism is the canonical comparison, since every step respects the limiting projections. This also covers the empty finite diagram and hence the terminal object. Thus $(iv)\Rightarrow (i)$, completing the cycle: