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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.

A preadditive category has an abelian-group structure on every hom-set and bilinear composition. One equivalent definition of an additive category is a preadditive category with a zero object and finite products in a category; the following argument proves that these products in a category are automatically coproducts in a category. An abelian category is an additive category with kernels and cokernels in which every monomorphism is a kernel and every epimorphism is a cokernel. Equivalently, for every arrow the canonical map from its coimage to its image is invertible.

Let $P={\prod }_{i=1}^n{A}_i$, with projections $p_i$. Define ${\iota }_i:A_i\to P$ by $p_j{\iota }_i={\delta }_{ji}$, where the diagonal entry is the identity and the others are zero. Since its composites with all projections are identities,Given $f_i:A_i\to B$, the map $f={\sum }_i{f}_i{p}_i$ satisfies $f{\iota }_i=f_i$. If another map has those composites, multiplying the displayed identity by that map forces it to equal $f$. Thus $P$ is also a coproduct in a category. The empty case is the zero object. Reversing arrows proves that a finite coproduct in a category is likewise a product in a category. Hence finite products in a category and coproducts in a category are canonically biproducts in an additive category.

For the category of abelian groups, let $(A_i,D_{ij})$ be a directed diagram, with no injectivity assumption on its transition maps. Its set colimit consists of classes $[i,a]$, withDirectedness proves transitivity of this relation. Define addition by moving representatives to a common upper index:Moving all indices involved to one further upper bound proves independence of representatives and of $k$. Define $-[i,a]=[i,-a]$ and use the common class of the zeros as zero. The abelian-group laws hold at a common index, so descend to the quotient. Every canonical map $A_i\to C$ is a homomorphism, and a compatible cocone induces the homomorphism $[i,a]\mapsto f_i(a)$. This is the unique extension. Moreover the formula for addition is forced by those canonical homomorphisms, proving that the underlying-set functor creates directed colimits. This establishes directed colimits created by the abelian-group forgetful functor explicitly.

If a cocone ${\nu }_i:A_i\to B$ has injective legs, the induced map $C\to B$ is injective. Indeed, equal images of $[i,a]$ and $[j,b]$ give, at a common index $k$,Injectivity of ${\nu }_k$ gives equal representatives at $k$. Since monomorphisms of abelian groups are exactly injective homomorphisms, we have proved that the category of abelian groups is a finitary abelian category in the directed-union sense of the question.

Now let $\mathcal{A}$ be complete, cocomplete and finitary. Index by finite subsets $J\subseteq I$ and put $S_J={⨁}_{j\in J}{A}_j$. For $J\subseteq K$, insert the additional zero coordinates. A compatible cocone on these finite biproducts is exactly a family of maps from every $A_i$, so their directed colimit is $S={\coprod }_{i\in I}{A}_i$.

Each finite biproduct has a map $k_J:S_J\to P={\prod }_{i\in I}{A}_i$ using its coordinates in $J$ and zeros outside $J$. Projection to the finite set of coordinates in $J$ is a retraction, so $k_J$ is a split monomorphism. These maps form a cocone of monomorphisms. The finitary property makes its induced map a monic arrow, and that induced map is exactly the canonical infinite identity matrix:This proves the coproduct-to-product comparison in a finitary abelian category. The PDF's use of finite products in a category in the hint is valid because these are the same finite biproducts.

If $\mathcal{A}$ is also a cofinitary abelian category, apply this argument in its opposite category. The same canonical comparison $j$ is then an epic arrow in $\mathcal{A}$. In an abelian category an arrow that is both a monomorphism and an epimorphism is invertible, so $j$ is an isomorphism.

For countably many copies of $A$, put $S={\coprod }_{n\ge 0}A$ and let $q=j^{-1}\Delta :A\to S$, where $\Delta$ is the product in a category diagonal. Let ${\iota }_n$ denote the coproduct in a category injections, let $s:S\to S$ shift them by $s{\iota }_n={\iota }_{n+1}$, and let $\nabla :S\to A$ be the codiagonal. Comparing every product in a category coordinate givesSince $j$ is a monic arrow, $q={\iota }_0+sq$. Also $\nabla {\iota }_0=1_A$ and $\nabla s=\nabla$. Therefore, for $z=\nabla q$,Hom-sets are abelian groups, so cancellation gives $1_A=0$. Any arrow to or from $A$ is consequently zero by composition with its identity. Thus every object is both initial and terminal, andThis is the countable shift-and-fold argument behind countable biproducts in an additive category force triviality.

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.

Take $A=1$. By assumption $E=(1\downarrow F)$ has finite limits, so $E^{\mathrm{op}}$ has finite colimits. A filtered category is nonempty, admits a cocone over every pair of objects, and admits a coequalizing arrow for every parallel pair. The initial object of $E^{\mathrm{op}}$ gives nonemptiness, its binary coproducts give the required two-object cocones, and its coequalizers equalize parallel arrows. Consequently $\boxed{\text{(ii)}\Rightarrow \text{(iii)}}$. These statements also handle the otherwise possible empty category of elements.

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.

Use 0-presentable object in the specified sense: its hom-functor preserves all small colimits. For a representable $H_c$, the Yoneda lemma identifies this hom-functor with evaluation at $c$. Given any small diagram $P_j$, pointwise colimits giveThese are the canonical comparison bijections, including for an empty diagram. Therefore every representable presheaf is 0-presentable, or a small projective object in this sense.

By the canonical colimit presentation of a presheaf, writewith structure maps ${\alpha }_{c,x}:H_c\to P$. Since $P$ is a 0-presentable object, the canonical mapis bijective. Every element of a set-valued colimit has a representative in some component. In particular $1_P$ is the image of a transformation $s:P\to H_c$ for some $(c,x)$, soThus $P$ is a retract of $H_c$. The representable retract criterion from part (c), using the assumed equalisers in $\mathcal{C}$, now gives $P$ is representable. Only existence of the representative of $1_P$ is needed; no stronger uniqueness-of-factorization assertion about individual colimit injections has been assumed.

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.

Let $F:\mathcal{C}\to \mathcal{D}$ be a final functor, and let ${\lambda }_c:GFc\to V$ be a cocone under a diagram $GF$. For $d\in \mathcal{D}$, choose an object $(c,a:d\to Fc)$ of the nonempty comma category $(d\downarrow F)$ and defineIf $t:(c,a)\to (c^{\prime },a^{\prime })$ is a morphism in that comma category, then $Fta=a^{\prime }$, so the cocone equation gives ${\lambda }_{c}G(a)={\lambda }_{c^{\prime }}G(Ft)G(a)={\lambda }_{c^{\prime }}G(a^{\prime })$. Equality also holds along a reversed arrow, and hence along any zigzag. Connectedness makes the definition independent of the chosen object. For $v:d\to d^{\prime }$, use $(c,a:d^{\prime }\to Fc)$ and $(c,av:d\to Fc)$ to obtain ${\bar{\lambda }}_d={\bar{\lambda }}_{d^{\prime }}Gv$. Thus $\bar{\lambda }$ is a cocone under $G$. Taking $(c,1_{Fc})$ shows that it extends $\lambda$.

Every extension must satisfy ${\bar{\lambda }}_d={\lambda }_{c}G(a)$ by its cocone equation, proving uniqueness. The construction commutes with postcomposition by a morphism $V\to V^{\prime }$. Hence restriction gives a natural bijection between the sets of cocones of $G$ and $GF$ with any fixed vertex. A universal cocone for $GF$ therefore extends to a universal cocone for $G$. In particular,whenever the latter exists, and existence of all colimits of shape $\mathcal{C}$ implies existence of all colimits of shape $\mathcal{D}$.

For the factorization of an arbitrary functor $H:\mathcal{C}\to \mathcal{E}$, letthe set of connected components of a category, where components are defined by finite zigzags. Precomposition by $v:e\to e^{\prime }$ gives a functor $(e^{\prime }\downarrow H)\to (e\downarrow H)$ and therefore a map $P(v):P(e^{\prime })\to P(e)$. Identity and composition laws make $P$ a categorical presheaf. This is a set-valued construction when the relevant comma categories are small or have only a set of components, as in the usual small-category convention for this factorization.

Take $\mathcal{D}$ to be the category of elements of this presheaf. Its objects are pairs $(e,S)$ with $S\in P(e)$; a morphism $(e,S)\to (e^{\prime },S^{\prime })$ is an arrow $v:e\to e^{\prime }$ satisfying $P(v)(S^{\prime })=S$. Define $G:\mathcal{D}\to \mathcal{E}$ by forgetting $S$. Given $v:e\to e^{\prime }$ and an object $(e^{\prime },S^{\prime })$, the unique lift into that object has source $(e,P(v)(S^{\prime }))$ and underlying arrow $v$. This proves that $G$ is a discrete fibration.

Define $F(c)=(Hc,[(c,1_{Hc})])$. For $t:c\to c^{\prime }$, take $F(t)=Ht$. The arrow $t$ in $(Hc\downarrow H)$ joins $(c,1_{Hc})$ to $(c^{\prime },Ht)$, so the required component equation holds. It follows that $F$ is a functor and $GF=H$.

Finally, an object of $((e,S)\downarrow F)$ is a pair $(c,v:e\to Hc)$ whose component in $(e\downarrow H)$ is $S$. Its morphisms are exactly the morphisms of that comma category between objects in $S$. Consequently $((e,S)\downarrow F)$ is precisely the full connected component $S$, and is nonempty and connected. Thus $F$ is final, and every such functor factors as a final functor followed by a discrete fibration.

Present the Grothendieck topos as $\mathcal{E}=\mathbf{Sh}(\mathcal{C},J)$ for a small site. Let $a:\hat{\mathcal{C}}\to \mathcal{E}$ be sheafification and $i$ the inclusion. The standard sheafification construction supplies $a⊣i$, with $a$ preserving finite limits. Concretely, matching-family constructions commute with finite limits, and their directed refinement over covering sieves does so as well. Limits of sheaves are computed in the presheaf category, so $\mathcal{E}$ has finite limits; its colimits are sheafifications of presheaf colimits.

For sheaves $A,B$, form the presheaf exponentialIt is a sheaf. Indeed, for a covering sieve $S\hookrightarrow yc$, one has $aS\cong a(yc)$. Exponential adjunction, sheafification adjunction and left exactness giveThese are the restriction comparisons, proving the sheaf condition. Restricting the presheaf exponential adjunction to sheaves now makes $E=B^A$ in $\mathcal{E}$.

The subobject classifier is the sheaf ${\Omega }_J$ of J-closed sieves. A sieve $S$ on $c$ is J-closed when, for every $f:d\to c$, $f^{\ast }S\in J(d)$ implies $f\in S$. Pullback of sieves defines its restrictions, and the maximal sieve defines truth. Local sieve data glue by taking the J-closure of their compatible generated sieve; uniqueness follows because membership in a J-closed sieve is local. This proves that ${\Omega }_J$ is a sheaf.

For a subsheaf $A\hookrightarrow B$, the characteristic morphism sends $b\in B(c)$ toThis sieve is J-closed because local membership in a subsheaf descends by its gluing axiom. Pulling truth back recovers exactly $A$. Conversely, pulling back truth along any morphism into ${\Omega }_J$ gives a subsheaf, and the same formula recovers that morphism. Finite limits, exponentials and this classifier make every Grothendieck topos an elementary topos.

Here are the explicit Heyting operations on subsheaves of an object $B$. Intersections give meets:The top is $B$. Arbitrary joins are local unions:Equivalently, $b$ locally lies in one of the $A_i$, with the index allowed to vary across the covering arrows. This is the J-closure of the pointwise union, or its sheafification. It is the smallest subsheaf containing every $A_i$, since any such subsheaf must contain sections locally in it.

The bottom is the empty join, namely the initial subobject. Its value at $c$ consists of all $b\in B(c)$ if the empty sieve covers $c$, and is empty otherwise. This detail matters on sites with empty covers; it is not safe to use an always-empty presheaf as the initial sheaf.

Implication is described without any pointwise-complement assumption:Restrictions preserve this condition. It is also local: pull a covering sieve back along an arbitrary $f$, use membership in $A$ on those restrictions, then descend their membership in $D$. Thus it is a subsheaf. It satisfies the defining Heyting algebra adjunctionFor the forward direction take the identity restriction. For the reverse direction every restriction of a section of $C$ remains in $C$, so membership in $A$ forces membership in $D$. Negation is $\neg A=A\Rightarrow 0$. These formulas give the complete Heyting algebra structure on ${\mathrm{Sub}}_{\mathcal{E}}(B)$; in general negation is a pseudocomplement rather than a set-theoretic complement.

For the covariant category in this part, put $\mathcal{E}=[\mathcal{C},\mathbf{Set}]$ and $r_c=\mathcal{C}(c,-)$. Its global-sections functor is $\Gamma =\mathrm{Hom}(1,-)$, the limit of the diagram. If $\Gamma$ is also an inverse image, it is a left adjoint and preserves all small colimits.

Consider the canonical pointwise-surjective mapAt $d$, the summand $r_d(d)$ contains $1_d$, so $q$ is an epimorphism. In a presheaf topos epimorphisms are regular, and a colimit-preserving functor preserves their coequalizer presentations. Thus $\Gamma q$ is surjective. Since $\Gamma$ preserves coproducts, some summand has a global element, yielding a section $s:1\to r_c$ of its unique map $r_c\to 1$. The constant singleton functor is a retract of a covariant representable.

Conversely, if $1$ is such a retract in a category, $\mathrm{Hom}(1,-)$ is a retract of $\mathrm{Hom}(r_c,-)$, which is evaluation at $c$ by the Yoneda lemma. Evaluation preserves colimits pointwise. The colimit comparison for a retract functor is a retract of the evaluation comparison; a retract of an isomorphism is an isomorphism, so $\Gamma$ preserves all small colimits. It also preserves finite limits because it is representable. A right adjoint can be constructed byFor $h:d\to d^{\prime }$, precomposition gives $r_{d^{\prime }}\to r_d$, and applying $\Gamma$ and then $\mathrm{Set}(-,S)$ gives $R(S)(d)\to R(S)(d^{\prime })$. The presentation of every functor as a colimit of representables gives $\mathrm{Hom}(F,R(S))\cong \mathrm{Set}(\Gamma F,S)$. Hence $\Gamma ⊣R$, so $\Gamma$ is an inverse image and the topos is local.

The retract condition has a precise interpretation as the initial object criterion for a covariant local topos. Write the section as a natural family $u_d:c\to d$. Naturality says $h{u}_d=u_{d^{\prime }}$ for every $h:d\to d^{\prime }$. Put $e=u_c$. Then $e^2=e$ and every $f:c\to d$ satisfies $fe=u_d$. In the idempotent completion of $\mathcal{C}$, the object $(c,e)$ is initial: an arrow from it to $(d,k)$ must satisfy $v=kve$, and the identities force $v=u_d$; this arrow exists because $k{u}_d=u_d$ and $u_{d}e=u_d$.

Conversely an initial object $(c,e)$ in the idempotent completion gives that natural family and the retract. ThereforeIf $\mathcal{C}$ is already idempotent-complete, this is equivalent to an initial object in $\mathcal{C}$ itself, and $\Gamma$ is evaluation there. The word is initial, not terminal, because this question uses covariant functors.

One cannot omit idempotent completion for a general small category. Take the one-object category of the monoid $\{1,0\}$ with absorbing zero. It has no initial object because its endomorphism set has two elements, but the natural family $u=0$ satisfies $m0=0$ for every $m$. Its covariant functor category is local. Concretely, a functor is a set with an idempotent operator, and its global sections are the fixed points of that operator.

For a topological space $X$, there is an equally explicit criterion. If $\mathbf{Sh}(X)$ is local, apply $\Gamma$ to the epimorphism ${\coprod }_i{U}_i\to 1$ associated with any open cover $X={\bigcup }_i{U}_i$, regarding the opens as subterminal sheaves. Since $\Gamma U_i$ is a singleton exactly when $U_i=X$, and otherwise empty, coproduct and epimorphism preservation force some $U_i=X$.

Thus every open cover of $X$ contains $X$ itself. Conversely this open cover criterion for a local sheaf topos condition is equivalent to existence of a point $x$ whose only open neighborhood is $X$: the union of all proper open sets cannot be all of $X$, so choose $x$ outside it. Such a point plainly forces any cover to contain $X$. Empty spaces fail the condition.

At this point, the stalk functor $F\mapsto F_x$ is simply $F(X)=\Gamma F$, since there is only one open neighborhood. Its right adjoint is the skyscraper sheaf of setsA morphism $F\to R(S)$ is determined exactly by its map $F(X)\to S$, because all proper opens omit $x$. Hence $\Gamma ⊣R$, and $\Gamma$ preserves finite limits, proving locality. ConsequentlyIn a $T_0$ space this distinguished point is unique and closed. In a $T_1$ space the condition forces $X$ to be a singleton. Indistinguishable points in a non-$T_0$ space can all have this property; uniqueness of an actual point is not part of the general criterion.