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The arrow category of $\mathcal{C}$ has morphisms $A_0\to A_1$ of $\mathcal{C}$ as objects and commutative squares as morphisms. It is the functor category $[2,\mathcal{C}]$, where $2$ is the category with one nonidentity arrow.

It is also the comma category $({\mathrm{id}}_{\mathcal{C}}\downarrow {\mathrm{id}}_{\mathcal{C}})$.

Objects are small categories and morphisms are functors. Its products pair objects and arrows componentwise. Its exponential objects are functor categories, as shown by the currying adjunction for small categories.

For a small category, this formula defines the exponential object in its covariant set-valued functor category. An arrow $c\to d$ acts by precomposition with $\mathcal{C}(d,-)\to \mathcal{C}(c,-)$. Evaluation applies a natural transformation to $(1_c,z)$; currying a map $P\times G\to H$ sends $p\in P(c)$ to the transformation $(f,z)\mapsto \theta (P(f)p,z)$. The formulas prove the universal property directly, without an adjoint existence theorem.

A functor creates categorical limits of a specified shape when every given limiting categorical cone over an image diagram uniquely lifts to a categorical cone over the original diagram, and the lift is limiting. This is stronger than just being a limit-reflecting functor. Forgetting arrows from a functor category creates its pointwise limits in a functor category.

An M-set is a set carrying a monoid action. This terminology here uses left actions. Morphisms are equivariant maps of monoid sets, and the resulting functor category $[M,\mathbf{Set}]$ is an elementary topos.

Let $F:\mathcal{C}\to \mathcal{D}$, $G:\mathcal{D}\to \mathcal{C}$, $\eta :1\Rightarrow GF$ and $\epsilon :FG\Rightarrow 1$ be natural transformations. If $G\epsilon {\eta }_G=1_G$, then $e={\epsilon }_{F}F\eta :F\Rightarrow F$ is an idempotent morphism in the functor category. Naturality implies $Ge\eta =\eta$ and $\epsilon e_G=\epsilon$; these absorption identities prove $e^2=e$. The omitted triangle measures exactly whether $e=1_F$.

Let $D:J\to [\mathcal{C},\mathcal{S}]$ be a small diagram in a category. Since $\mathcal{S}$ is a complete category, choose at each $C\in \mathcal{C}$ a categorical limitFor $f:C\to C^{\prime }$, the arrows $D_j(f)p_{j,C}$ form a cone over a diagram at $C^{\prime }$, because the diagram arrows $D_j\to D_k$ are natural transformations. There is therefore a unique arrow $L(f):L(C)\to L(C^{\prime })$ withThe universal property immediately gives $L(1_C)=1_{L(C)}$ and $L(gf)=L(g)L(f)$: both sides have the same composites with every limit projection. Thus $L$ is a functor, and the displayed equations make the $p_j$ natural transformations.

Given any categorical cone $q_j:X\to D_j$ in the functor category, its components induce unique $q_C:X(C)\to L(C)$. For $f:C\to C^{\prime }$, compare $L(f)q_C$ and $q_{C^{\prime }}X(f)$ after each $p_{j,C^{\prime }}$; naturality of $q_j$ makes their composites equal. Hence $q$ is a natural transformation, and uniqueness is componentwise. This proves that $L$ is a categorical limit in $[\mathcal{C},\mathcal{S}]$.

The functor category is complete, and every evaluation functor preserves limits, since evaluating the constructed limit at $C$ gives exactly the chosen limit in $\mathcal{S}$. This is the pointwise limits in a functor category construction; it also covers the empty diagram and its terminal object.

For $c\in \mathcal{C}$ and $d\in \mathcal{D}$, define maps of hom-setsThey are natural in $c,d$, using naturality of $\eta ,\epsilon$. The one supplied triangle identity for an adjunction givesNaturality of $\epsilon$ also givesSince $\alpha \beta =1$, the composite $\beta \alpha$ is idempotent. Evaluating its equation $(\beta \alpha {)}^2=\beta \alpha$ on $f=1_{Fc}$ gives $e_c^2=e_c$. As $e$ is a composite of natural transformations, this proves that $e$ is an idempotent in the functor category, including the required naturality.

Suppose this one-triangle adjunction idempotent splits as natural transformations $F\stackrel{r}{\to }H\stackrel{i}{\to }F$ with $ri=1_H$ and $ir=e$. The maps $h\mapsto hr$ and $f\mapsto fi$ give inverse bijections between $\mathcal{D}(Hc,d)$ and the subset of $\mathcal{D}(Fc,d)$ consisting of maps with $f{e}_c=f$. Indeed $(hr)e_c=hrir=hr$, $(hr)i=h$, and $(fi)r=f{e}_c=f$. The maps $\alpha ,\beta$ give inverse bijections between that subset and $\mathcal{C}(c,Gd)$, since $\beta \alpha (f)=f{e}_c$ and $\alpha \beta =1$. Combining them gives the natural bijectionwith inverse $g\mapsto {\epsilon }_{d}F(g)i_c$. Hence $H⊣G$. Its adjunction unit is $Gr\eta$ and its adjunction counit is $\epsilon i_G$.

Conversely, suppose $H⊣G$, with natural bijection $\theta :\mathcal{D}(Hc,d)\cong \mathcal{C}(c,Gd)$. Set $\tau ={\theta }^{-1}\alpha$ and $\sigma =\beta \theta$. Then $\tau \sigma =1$ and $\sigma \tau (f)=f{e}_c$. By the covariant Yoneda lemma, the transformation $\tau :\mathcal{D}(Fc,-)\to \mathcal{D}(Hc,-)$ is precomposition with a unique morphism $i_c:Hc\to Fc$, and $\sigma$ is precomposition with a unique morphism $r_c:Fc\to Hc$. The equation $\tau \sigma =1$ evaluated at $1_{Hc}$ gives $r_c{i}_c=1_{Hc}$; the equation $\sigma \tau (f)=f{e}_c$ evaluated at $1_{Fc}$ gives $i_c{r}_c=e_c$. Naturality in $c$ of $\tau ,\sigma$ implies, again by the same Yoneda lemma, that $i,r$ are natural transformations. Thus they split $e$ in the whole functor category, rather than merely at individual objects. We have proved

Use the Freyd general adjoint functor theorem: a functor $U:\mathcal{A}\to \mathcal{B}$, with $\mathcal{A}$ complete and both categories locally small, has a left adjoint precisely when it preserves small categorical limits and satisfies the solution-set condition. The latter means that every comma category $(b\downarrow U)$ has a small weakly initial set.

Here is the proof. First prove the initial-object lemma for complete categories with a weakly initial set. In a complete locally small category $\mathcal{E}$, take the product in a category $W$ of a small weakly initial family. It is weakly initial: project to whichever member admits a morphism into the required object. Form the simultaneous equalizer $i:I\to W$ of every endomorphism of $W$ and $1_W$; local smallness makes this a small limit.

For parallel $f,g:I\to X$, let $j:Y\to I$ be their equalizer. Weak initiality supplies $t:W\to Y$. Since $ijt$ is an endomorphism of $W$, the defining equalities give $ijti=i$. Cancel the monomorphism $i$ to get $jti=1_I$. Thus $j$ is a split epimorphism as well as a monomorphism, hence invertible, and $f=g$. Existence of a morphism $I\to X$ follows by composing $i$ with a map $W\to X$. Therefore $I$ is an initial object.

Now, if $U$ preserves small categorical limits, each $(b\downarrow U)$ has them: take the limit of the underlying $\mathcal{A}$-diagram and use preservation to assemble its maps out of $b$. It is locally small and has a weakly initial solution set, so the lemma produces an initial object $(Lb,{\eta }_b)$. Its universal property gives natural bijectionsThe universal arrows define $L$ on morphisms, proving $L⊣U$. Conversely, a right adjoint preserves categorical limits, and its unit supplies a singleton solution set in each comma category.

Apply this to $U=F^{\ast }$. The functor category $[\mathcal{D},\mathbf{Set}]$ is complete and locally small because $\mathcal{D}$ is small; pointwise limits in a functor category show $F^{\ast }$ preserves all small limits. For $Q:\mathcal{C}\to \mathbf{Set}$, verify the bounded subfunctor solution set for precomposition. Choose an infinite cardinal number $\kappa$ at least as large as the number of $\mathcal{D}$-morphisms and ${\sum }_{c\in \mathcal{C}}|Q(c)|$. Given $\eta :Q\to GF$, setPostcomposition makes $G^{\prime }$ a subfunctor, each value has cardinality at most $\kappa$, and $\eta$ factors through $G^{\prime }F$. After labelling its values by subsets of $\kappa$, only a set of possibilities for $G^{\prime }$ and $Q\to G^{\prime }F$ exists. Their maps into $G$ give the required weakly initial set. Thus $F^{\ast }$ has a left adjoint.

It is the left Kan extension. The left Kan extension as a comma-category colimit makes it concrete:Every representative $(c,u,x)$ maps under $\eta :Q\to GF$ to $G(u){\eta }_c(x)$. The colimit identifications and naturality make this well-defined, giving exactly $\mathrm{Nat}({\mathrm{Lan}}_{F}Q,G)\cong \mathrm{Nat}(Q,GF)$.

An essay arguing against the claim. A good definition can change which mathematical questions are visible. Nevertheless, I would argue that category theory derives its explanatory force from the interaction between definitions and theorems; there is no sound general ordering in which definitions contribute more than theorems.

The strongest case for the contrary view comes from universal properties. A product in a category is characterized by maps into it, rather than by a particular construction of ordered pairs. The same definition applies to sets, groups, and topological spaces. An adjunction goes further: its natural correspondence of arrows puts free constructions, quotient constructions, and several kinds of completion into a common language. Such definitions remove irrelevant choices and expose relationships that would be difficult to recognize from separate constructions.

But a universal-property definition does not supply an object. The definition of a product in a category says what a product would accomplish; an existence theorem says whether the category actually has one. Even uniqueness up to unique isomorphism is a short proof, not an extra clause to be silently supplied by the word “universal”. Given two products, their universal properties produce maps in both directions, and uniqueness forces the two composites to be identities. This elementary theorem makes the definition usable independently of a chosen construction.

The Yoneda lemma illustrates the distinction especially clearly. A representable functor is a definition, but the lemma proves that its natural transformations into any set-valued functor are determined by one element. Its proof evaluates at the identity and reconstructs the whole transformation by functoriality. Consequently the Yoneda embedding is full and faithful: an object's pattern of incoming arrows recovers its morphisms without loss. That is a substantive explanation of why the chosen definition is powerful. Merely calling a functor representable would not establish it.

Similarly, defining a monadic adjunction gives a precise meaning to recovering structured objects as algebras for a monad. It does not decide whether a given forgetful functor is monadic. The precise monadicity theorem translates that difficult classification into a usable test involving adjoint functors, reflected isomorphisms, and specific coequalizers. For compact Hausdorff spaces, the crucial step is proving that a set-split relation is closed. The definition of a monad alone does not contain this topological argument. The theorem connects the algebraic language to an independently meaningful class of spaces.

There is also a danger in praising definitions without hypotheses. A Cartesian closed category has exponential objects, but a category of structured objects need not be cartesian closed merely because underlying sets have function sets. The correct exponential in a functor category generally involves natural transformations out of representable functors, not pointwise function sets. Its existence and universal property require an argument. Theorems also reveal when tempting definitions cannot support the intended construction.

Definitions provide the vocabulary and select the structures worth studying; proofs establish existence, consequences, and limitations. Category theory makes this partnership unusually explicit because its definitions are often expressed through universal properties. Its distinctive achievement is to turn well-chosen definitions into reusable theorems, rather than to make theorems secondary.

An alternative essay arguing for the claim. In category theory, theorems are indispensable, but a carefully chosen definition often performs the greater act of organization. It identifies what many apparently unrelated constructions have in common and makes whole families of proofs possible. In that explanatory sense, definitions can be the subject's most influential contributions.

Consider a universal property. Describing a product in a category as an object representing pairs of incoming arrows replaces several different concrete constructions by one invariant pattern. The corresponding uniqueness proof is important but short; the lasting insight is that the same pattern is the right question in every category. Once the definition is available, it becomes meaningful to ask which functors preserve products, whether the products exist in a new category, and how they interact with other universal constructions.

An adjunction is an even more striking example. The existence of free groups, free modules, and a Stone-Čech compactification can be proved separately. The definition of an adjunction recognizes their common relation to forgetful functors and explains why arrows out of a free object are determined by arrows out of its generators. It also packages the relation as a monad. A large collection of isolated existence results becomes an instance of a shared structure, with reusable consequences about composition and categorical limits.

A valuable definition also has to distinguish structures rather than merely rename them. An abelian category is not just an additive category with categorical kernels and categorical cokernels: normality of monomorphisms and epimorphisms matters. The additive example with truncated primary torsion shows why those clauses belong in the definition. The definition therefore preserves exactly the structure on which theorems about images and exact sequences depend.

This view does not make proofs dispensable. The Yoneda lemma justifies the use of representable functors, and the precise monadicity theorem tests when a proposed algebraic description is valid. Rather, it argues that the best categorical definitions have unusually broad explanatory reach: they select the invariant structures to which those proofs apply. If “matter more” means providing the reusable framework in which many theorems become instances of one idea, category-theoretic definitions have a strong claim to that role.

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.

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.

For the covariant version of the Yoneda embedding, put $Y(A)=\mathcal{C}(A,-)$. A morphism $u:A\to B$ induces the natural transformation $Y(u):Y(B)\to Y(A)$ whose component sends $v:B\to X$ to $vu$. Thus $Y$ is a functor from the opposite category ${\mathcal{C}}^{\mathrm{op}}$ to the functor category $[\mathcal{C},\mathbf{Set}]$. The locally small category hypothesis makes each value a set.

The Yoneda lemma says that, for a functor $F:\mathcal{C}\to \mathbf{Set}$, evaluation at the identity morphism gives a natural bijectionGiven $a\in F(A)$, define ${\theta }_X^a(v)=F(v)(a)$. For $w:X\to X^{\prime }$, the functor law gives $F(w){\theta }_X^a(v)=F(wv)(a)={\theta }_{X^{\prime }}^a(wv)$, proving naturality. Conversely, naturality of $\theta$ at $v:A\to X$ gives ${\theta }_X(v)=F(v){\theta }_A(1_A)$. These two constructions are inverse, because $F(1_A)(a)=a$.

The bijection is natural in both variables. A natural transformation $\beta :F\to H$ sends the element $a$ to ${\beta }_A(a)$, matching postcomposition by $\beta$. For $u:A\to B$, precomposing with $Y(u)$ sends $\theta$ to a transformation from $Y(B)$ and its distinguished element is ${\theta }_B(u)=F(u)(a)$. This is exactly the required dependence on $A$.

Taking $F=Y(B)$ gives $\mathrm{Nat}(Y(A),Y(B))\cong \mathcal{C}(B,A)$. Consequently the covariant Yoneda embedding is full and faithful, with the reversal of arrows accounted for by its opposite category domain.

Suppose every component ${\alpha }_A$ is a monomorphism in the Category of sets, hence an injective function. If $\beta ,\gamma :H\to F$ are natural transformations with $\alpha \beta =\alpha \gamma$, then ${\alpha }_A{\beta }_A={\alpha }_A{\gamma }_A$ at every object. Componentwise cancellation gives ${\beta }_A={\gamma }_A$, so $\beta =\gamma$. Thus $\alpha$ is a monomorphism in the functor category.

Conversely, suppose $\alpha$ is a monomorphism and $x,y\in F(A)$ satisfy ${\alpha }_A(x)={\alpha }_A(y)$. The Yoneda lemma supplies natural transformations ${\theta }^x,{\theta }^y:Y(A)\to F$. Its naturality in $F$ identifies $\alpha {\theta }^x$ and $\alpha {\theta }^y$ with the equal elements ${\alpha }_A(x)$ and ${\alpha }_A(y)$. Cancellation by $\alpha$ gives ${\theta }^x={\theta }^y$; evaluating at $1_A$ gives $x=y$. Therefore a natural transformation is a monomorphism exactly when all its components are injective functions. This is the pointwise monomorphism in a set-valued functor category criterion.

Construct pointwise limits in a functor category. For $H:\mathcal{J}\to [\mathcal{D},\mathcal{C}]$, choose at each object $d$ a categorical limit $L(d)$ of $j\mapsto H_j(d)$, with projections $p_j(d)$. For $u:d\to d^{\prime }$, the family $H_j(u)p_j(d)$ is compatible; define $L(u)$ uniquely byThe universal property gives $L(1_d)=1_{L(d)}$ and $L(vu)=L(v)L(u)$, since those equalities hold after every projection. Thus $L$ is a functor, and each $p_j:L\Rightarrow H_j$ is a natural transformation.

For any categorical cone $(M,(a_j))$, the pointwise categorical limits give unique maps $a(d):M(d)\to L(d)$. To check naturality, compose $L(u)a(d)$ and $a(d^{\prime })M(u)$ with every $p_j(d^{\prime })$; both become $H_j(u)a_j(d)=a_j(d^{\prime })M(u)$. The projections distinguish arrows into their categorical limit, so the two maps agree. Componentwise uniqueness gives uniqueness of the natural transformation $a$. This proves that $(L,p_j)$ really is the required categorical limit, rather than merely a family of objectwise candidates.

A specified categorical limit categorical cone in ${\mathcal{C}}^{\mathrm{ob}\mathcal{D}}$ fixes these objectwise vertices and projections. The displayed equation forces every arrow $L(u)$, and the argument forces every categorical cone factorization. Hence the forgetful functor uniquely lifts that categorical cone and is a limit-creating functor. The construction only takes small categorical limits in $\mathcal{C}$; it does not require $\mathcal{D}$ to be small. As usual, the functor categories are understood in a universe where their collections of transformations are meaningful.

The two introductory requests can be settled before the lettered applications. In the functor category $[C,\mathbf{Set}]$, finite limits and finite unions of subobjects are pointwise. The component of the diagonal at $c$ is ordinary equality on $F(c)$. Its only possible complement isThis forms a subfunctor exactly when each $F(u)$ sends unequal elements to unequal elements, equivalently when every $F(u)$ is injective. In that case $D$ and the diagonal are disjoint and their union is $F^2$ at every component. Conversely, a diagonal complement must have these components and be stable under every transition map. Hence $F$ is decidable exactly when every transition map is injective.

Regard a monoid $M$ as a one-object category; a covariant set-valued functor is a left M-set. Give $M\times A$ the diagonal left action $t\cdot (w,a)=(tw,t\cdot a)$. LetRight multiplication on the first coordinate commutes with the diagonal left action, so $m\cdot f$ remains equivariant. The formula obeys $m\cdot (n\cdot f)=(mn)\cdot f$ and $1\cdot f=f$. Evaluation isIt is equivariant because $\mathrm{ev}(m\cdot f,m\cdot a)=f(m,m\cdot a)=m\cdot f(1,a)$.

For an equivariant $h:C\times A\to B$, defineThis is equivariant in $(w,a)$, and $\hat{h}(m\cdot c)=m\cdot \hat{h}(c)$. Evaluation recovers $h$. Conversely, currying the evaluation of a map $C\to E$ recovers that map by its equivariance. This proves the exponential universal property and the natural identificationwith exactly the stated action. In particular, decidability in a set-valued functor category says that a left M-set is decidable if and only if each of its action maps is injective.

In the Category of sets, an epimorphism is exactly a surjective function. A surjective function is right-cancellable. If $q:X\to Y$ misses $y\in Y$, the constant-zero function and the function that is one at $y$ and zero elsewhere are distinct maps $Y\to \{0,1\}$ with equal composites with $q$.

If every ${\alpha }_A$ is surjective, equality $\beta \alpha =\gamma \alpha$ for natural transformations $\beta ,\gamma :G\Rightarrow H$ gives ${\beta }_A={\gamma }_A$ for every $A$. Thus $\alpha$ is an epimorphism in the functor category.

For the converse, construct the pointwise amalgamated doublewhere exactly the two copies of each element of $\mathrm{im}{\alpha }_A$ are identified; different elements of $G(A)$ remain different. Define $H(u)[y,i]=[G(u)y,i]$. Naturality of $\alpha$ implies that $G(u)$ takes its image into the image at the target, so this formula is well-defined and gives a functor. The maps $j_{i,A}(y)=[y,i]$ form natural transformations $j_0,j_1:G\Rightarrow H$, with $j_0\alpha =j_1\alpha$.

If $\alpha$ is an epimorphism, $j_0=j_1$. Since the two copies of an element outside $\mathrm{im}{\alpha }_A$ would be distinct, every element must lie in that image. Hence $\alpha$ is epic if and only if every component is epic. This proves the pointwise epimorphism in a functor category criterion directly, including the needed existence and naturality of the separating functor.

Take an epimorphism $\alpha :F\Rightarrow G$ in $[\mathcal{C},\mathbf{Set}]$ and a natural transformation $\beta :\mathcal{C}(A,-)\Rightarrow G$. By the Yoneda lemma, $\beta$ corresponds to $x={\beta }_A(1_A)\in G(A)$. By the pointwise epimorphism in a functor category criterion, choose $y\in F(A)$ with ${\alpha }_A(y)=x$.

The Yoneda lemma gives ${\gamma }_B(f)=F(f)(y)$, a natural transformation $\mathcal{C}(A,-)\Rightarrow F$. Naturality of $\alpha$ yieldsTherefore covariant representables are projective in the set-valued functor category. Local smallness ensures that $\mathcal{C}(A,-)$ is set-valued.