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Let $a:X\to Y$ be a natural transformation of categorical presheaves. If every $a_C:X(C)\to Y(C)$ is a surjective function, and $r,s:Y\rightrightarrows Z$ satisfy $ra=sa$, then $r_C{a}_C=s_C{a}_C$ at every $C$. Surjectivity gives $r_C=s_C$ for all $C$, hence $r=s$. Thus $a$ is an epimorphism.

Conversely suppose $y\in Y(C)$ is outside the image of some $a_C$. Construct the pushout witness for failure of pointwise surjectivity explicitly: at each $D$ takeRestrictions send $[(z,k)]$ to $[(Y(f)z,k)]$; naturality of $a$ ensures that every generating identification is respected. Thus $P$ is a categorical presheaf, and the two inclusions $j_0,j_1:Y\to P$ are natural transformations with $j_{0}a=j_{1}a$.

All identifications concern the same element $z$ in its two copies, so an element outside the image is never identified with its other copy. Consequently $j_{0,C}(y)\ne j_{1,C}(y)$, proving $a$ is not an epimorphism. ThereforeThis is the pointwise epimorphism in a functor category criterion. It requires neither a componentwise choice of preimages nor a natural section.

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.

A Cartesian closed category has finite products in a category and, for all $X,Y$, an exponential object $Y^X$ with a natural bijectionEquivalently the functor $-\times X$ has a right adjoint. Its counit of an adjunction is the evaluation map of an exponential object $Y^X\times X\to Y$.

For the presheaf category, define the exponential in a presheaf category byFor $f:D\to C$, its restriction sends $\alpha$ to $\alpha \circ (h_f\times 1_X)$. Composition of these restriction maps follows from composition of the $h_f$, so this is a categorical presheaf. Define evaluation map of an exponential object byFor $f:D\to C$, naturality of $\alpha$ givesThus $e$ is a natural transformation.

Given a natural transformation $\beta :Z\times X\to Y$, define its currying byNaturality of $\beta$ proves that ${\hat{\beta }}_C(z)$ is a natural transformation $h_C\times X\to Y$. For $f:C^{\prime }\to C$, the same formula proves that $\hat{\beta }:Z\to Y^X$ is natural. Evaluating at $g=1_C$ gives $e(\hat{\beta }\times 1_X)=\beta$.

Conversely, if $\gamma :Z\to Y^X$ and $\beta =e(\gamma \times 1_X)$, thenby naturality of $\gamma$. Hence the two constructions are inverse and natural in $Z,Y$. Products are available pointwise, including the terminal singleton presheaf, so the presheaf category is Cartesian closed. The exponential uses natural transformations out of $h_C\times X$; simply exponentiating the individual values generally does not give this object.

A monad on $\mathcal{C}$ consists of an endofunctor $T$ and natural transformations $\eta :1\to T$, $\mu :T^2\to T$ satisfying the unit and multiplication of a monad lawsAn algebra for a monad is an object $A$ equipped with $a:TA\to A$ such thatA morphism of algebras for a monad $f:(A,a)\to (B,b)$ is a morphism $f:A\to B$ satisfying $fa=bT(f)$. Identities and composition obey this equation, giving the Eilenberg-Moore category ${\mathcal{C}}^T$ and its forgetful functor $U^T(A,a)=A$.

For an adjunction $F⊣U:\mathcal{D}\to \mathcal{C}$ with induced monad $T=UF$, the Eilenberg-Moore comparison functor isThe functor $U$ is monadic when it has such a left adjoint and this comparison is an equivalence of categories; a stricter convention requires an isomorphism of categories over $\mathcal{C}$. We will identify which limit-lifting conclusion each convention supports in part (iii).

A monad on $\mathcal{C}$ consists of an endofunctor $T$ and natural transformations $\eta :1\to T$, $\mu :T^2\to T$ satisfying, at every object $A$,These are the associativity and unit laws. An algebra for a monad is $(A,a:TA\to A)$ withA monad algebra morphism $h:(A,a)\to (B,b)$ is a morphism $h:A\to B$ satisfying $ha=bTh$. Identity morphisms satisfy this equation, and if $h,k$ satisfy it, then $kha=kbTh=cTkTh=cT(kh)$. Hence these objects and morphisms form the Eilenberg-Moore category ${\mathcal{C}}^T$, with the forgetful functor $U(A,a)=A$ and $Uh=h$.

If $f:A\to B$ is a monomorphism, each function $\mathcal{C}(U,A)\to \mathcal{C}(U,B)$ given by postcomposition with $f$ is injective. Consequently, if two natural transformations into $H_A$ become equal after $H_{\bullet }(f)$, their components are equal, and the transformations themselves are equal. This proves that $H_{\bullet }(f)$ is a monomorphism in the presheaf category.

Conversely, suppose $H_{\bullet }(f)$ is a monomorphism and $u,v:U\to A$ satisfy $fu=fv$. Then $H_{\bullet }(f)H_{\bullet }(u)=H_{\bullet }(f)H_{\bullet }(v)$. Cancellation gives $H_{\bullet }(u)=H_{\bullet }(v)$, and faithfulness of the Yoneda embedding gives $u=v$. Hence $H_{\bullet }(f)$ is monic exactly when $f$ is monic.

Define the Yoneda embedding $H_{\bullet }:\mathcal{C}\to [{\mathcal{C}}^{\mathrm{op}},\mathbf{Set}]$ by $H_A(U)=\mathcal{C}(U,A)$ and $H_A(k)(h)=hk$. For $f:A\to B$, define $H_{\bullet }(f{)}_U(h)=fh$. Associativity proves naturality in $U$, and preservation of identity morphisms and composition in a category proves that $H_{\bullet }$ is a functor.

Apply the Yoneda lemma to $X=H_B$. It givesThe natural transformation corresponding to $f$ has component $h\mapsto fh$, exactly $H_{\bullet }(f)$. Therefore the induced map on every hom-set is a bijection: the Yoneda embedding is full and faithful.

For a categorical presheaf $X:{\mathcal{C}}^{\mathrm{op}}\to \mathbf{Set}$ and $A\in \mathcal{C}$, the Yoneda lemma gives a natural bijectionTo construct its inverse, take $x\in X(A)$ and set ${\theta }_U^x(h)=X(h)x$ for $h:U\to A$. For $k:V\to U$, the functor law gives $X(k){\theta }_U^x(h)=X(hk)x={\theta }_V^x(hk)$, so ${\theta }^x$ is a natural transformation. Conversely, naturality of any $\theta$ at $h:U\to A$ gives ${\theta }_U(h)=X(h){\theta }_A(1_A)$. Thus the two constructions are inverse, proving the bijection.

The bijection is natural in $X$: a natural transformation $t:X\to Y$ sends the distinguished element $x$ to $t_A(x)$ on either side. It is natural in $A$: for $f:A\to B$, precomposing a transformation $\mathcal{C}(-,B)\to X$ with postcomposition by $f$ sends its distinguished element $x\in X(B)$ to $X(f)x$. These equations prove the claimed naturality, rather than just an objectwise correspondence. Reversing morphisms gives the covariant form $\mathrm{Nat}(\mathcal{C}(A,-),Z)\cong Z(A)$ for $Z:\mathcal{C}\to \mathbf{Set}$.

We first justify the relevant pointwise epimorphism in a functor category criterion. A componentwise surjective function gives an epimorphism, because equality of two composites can be cancelled at each component. Conversely, for a natural transformation $t:X\to Y$, form the pointwise pushout $P=Y{⨿}_{X}Y$. Restrictions of $Y$ preserve the identifications imposed by $t$, so these pointwise sets and restrictions form a categorical presheaf. The two inclusions $j_1,j_2:Y\to P$ satisfy $j_{1}t=j_{2}t$. If $t$ is epic, $j_1=j_2$. At an object $U$, an element outside the image of $t_U$ would retain two distinct copies in $P(U)$, contradicting this equality. Thus every $t_U$ is surjective.

Apply this criterion to $H_{\bullet }(f)$ at $B$. Surjectivity of $\mathcal{C}(B,A)\to \mathcal{C}(B,B)$ supplies $g:B\to A$ with $fg=1_B$. Conversely, if $fg=1_B$, then $H_{\bullet }(f)H_{\bullet }(g)=1_{H_B}$, making $H_{\bullet }(f)$ a split epimorphism and hence an epimorphism. Therefore

The category of elements $\int X$ has objects $(W,x)$ for $x\in X(W)$, and morphisms $f:(W,x)\to (V,y)$ with $X(f)y=x$. Send $(W,x)$ to the representable presheaf $H_W$, and a morphism $f$ to $H_{\bullet }(f)$. The element $x$ supplies a natural transformation $H_W\to X$ with component $h\mapsto X(h)x$, by the Yoneda lemma. These transformations form a cocone.

Pointwise at $U$, the colimit of this diagram in a category has representatives $(W,x,h:U\to W)$, with relation $(W,X(f)y,h)\sim (V,y,fh)$. These are exactly the relations of the preceding categorical coend. Its proved natural isomorphism with $X(U)$ therefore identifies the cocone with the universal colimit cocone. More directly, every representative equals $(U,X(h)x,1_U)$ through the morphism $h:(U,X(h)x)\to (W,x)$; two representatives with the same image therefore agree. As $\mathcal{C}$ is small, $\int X$ is small, so this is an ordinary small colimit. Consequently

Working with a small category $\mathcal{C}$ (or in a fixed ambient universe), for a categorical presheaf $X$ and fixed $U$, apply the categorical coend construction to $T(A,B)=\mathcal{C}(U,B)\times X(A)$. It is the quotient of the disjoint union of pairs $(h:U\to W,x\in X(W))$ by the relationsDefine ${\Phi }_U([h,x])=X(h)x$. It respects every relation because $X(fh)x=X(h)X(f)x$. Define ${\Psi }_U(y)=[1_U,y]$. Then ${\Phi }_U{\Psi }_U(y)=y$. Conversely the relation with $f=h:U\to W$ gives $[h,x]=[1_U,X(h)x]$, so ${\Psi }_U{\Phi }_U$ is the identity morphism. ThereforeFor $k:V\to U$, restriction on the categorical coend sends $[h,x]$ to $[hk,x]$, and ${\Phi }_V([hk,x])=X(k){\Phi }_U([h,x])$. A natural transformation $X\to Y$ likewise acts on the second coordinate and commutes with $\Phi$. Thus the density formula for presheaves is a natural isomorphism, not just a family of unrelated bijections.

Given the two natural transformations, defineNaturality of $\eta$ and the second triangle identity for an adjunction giveNaturality of $\epsilon$ and the first triangle identity for an adjunction giveThe maps are natural in $C,D$, since the defining compositions commute with precomposition in $C$ and postcomposition in $D$ by naturality of $\eta ,\epsilon$. They therefore define an adjunction $F⊣G$. Their values on identity morphisms recover precisely $\eta ,\epsilon$. Conversely any adjunction with these unit and counit of an adjunction must have the same transpose formulas, so it is unique with the prescribed unit and counit.

Write the adjunction as a natural bijection ${\alpha }_{C,D}:\mathcal{D}(FC,D)\to \mathcal{C}(C,GD)$. The adjunction unit and adjunction counit are defined byNaturality of $\alpha$ implies $GF(u){\eta }_C={\eta }_{C^{\prime }}u$ for $u:C\to C^{\prime }$ and $v{\epsilon }_D={\epsilon }_{D^{\prime }}FG(v)$ for $v:D\to D^{\prime }$, so these are natural transformations $1_{\mathcal{C}}\to GF$ and $FG\to 1_{\mathcal{D}}$. More explicitly, naturality at $f:FC\to D$ and $g:C\to GD$ gives the two transpose formulasThese formulas will also prove the triangle identities for an adjunction.

For a presheaf of sets on a topological space $P$, define $LP=P(\varnothing )$ and $RP=P(S)$. Then the adjoints to constant presheaves on open sets areTo prove the first adjunction, a natural transformation $a:P\to \Delta A$ must satisfy $a_U=a_{\varnothing }\circ {\mathrm{res}}_{U,\varnothing }$ by naturality at the inclusion $\varnothing \subseteq U$. Hence it is uniquely determined by a function $P(\varnothing )\to A$. Every such function defines a natural transformation by this formula, because restriction maps compose. This gives $\mathrm{Nat}(P,\Delta A)\cong \mathbf{Set}(P(\varnothing ),A)$.

For the second adjunction, a natural transformation $b:\Delta A\to P$ satisfies $b_U={\mathrm{res}}_{S,U}\circ b_S$, so it is uniquely determined by a function $A\to P(S)$. Every such function defines the remaining components by restriction, giving $\mathrm{Nat}(\Delta A,P)\cong \mathbf{Set}(A,P(S))$. Evaluation acts on presheaf morphisms componentwise, so these object assignments are indeed functors. The use of $P(\varnothing )$ is essential: a constant presheaf of sets has value $A$ there, whereas a sheaf has a singleton value there. No sheaf condition is imposed in this question.