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For a locally small category $\mathcal{C}$, an object $A$, and a functor $F:\mathcal{C}\to \mathbf{Set}$, the covariant form of the Yoneda lemma is the natural bijectionFor $x\in F(A)$ define ${\alpha }_B^x(f)=F(f)(x)$, where $f:A\to B$. For $u:B\to D$, functoriality gives $F(u){\alpha }_B^x(f)=F(uf)(x)={\alpha }_D^x(uf)$, so this is a natural transformation. Conversely, naturality of $\alpha$ at $f:A\to B$ gives ${\alpha }_B(f)=F(f){\alpha }_A(1_A)$. The two constructions are inverse. They are natural in $F$, because a natural transformation $F\Rightarrow G$ commutes with evaluation. They are also natural in $A$: an arrow $v:A^{\prime }\to A$ induces $\mathcal{C}(A,-)\to \mathcal{C}(A^{\prime },-)$ by precomposition, and the resulting map $\mathrm{Nat}(\mathcal{C}(A^{\prime },-),F)\to \mathrm{Nat}(\mathcal{C}(A,-),F)$ corresponds to $F(v):F(A^{\prime })\to F(A)$. Applying the result to the opposite category gives $\mathrm{Nat}(\mathcal{C}(-,A),H)\cong H(A)$ for $H:{\mathcal{C}}^{\mathrm{op}}\to \mathbf{Set}$.

Now let $\mathcal{C}$ be a small category, and let $F,G:\mathcal{C}\to \mathbf{Set}$. Finite products in a category in $[\mathcal{C},\mathbf{Set}]$ exist pointwise; the terminal functor is the constant singleton functor. Define the exponential of covariant set-valued functors bySmallness ensures that these natural transformations form a set. For $u:C\to D$, define $(G^F)(u)(\alpha )$ at $E$ byPrecomposition respects identities and composition, so this defines a functor. Its evaluation map of an exponential object isTo check naturality, for $u:C\to D$ both sides give ${\alpha }_D(u,F(u)x)$: one side uses naturality of $\alpha$, and the other uses the defining action of $G^F$.

Here is the full universal property. Given $\theta :H\times F\Rightarrow G$, define $\hat{\theta }:H\Rightarrow G^F$ byFor $w:D\to E$, naturality of $\theta$ makes this commute with $G(w)$, so ${\hat{\theta }}_C(h)$ is a natural transformation. Replacing $h$ by $H(u)h$ is the same as precomposing $v$ by $u$, proving naturality in $C$. Conversely, $\beta :H\Rightarrow G^F$ gives ${\theta }_C(h,x)={\beta }_C(h{)}_C(1_C,x)$. Starting with $\theta$ recovers it immediately. Starting with $\beta$, its naturality at $v:C\to D$ yieldsso it too is recovered. The formulas are natural in $H$ and $G$, giving the required adjunction $(-)\times F⊣(-{)}^F$. Therefore the covariant functor category is cartesian closed. This proof supplies the actual exponential and does not need a colimit presentation of its arguments.

Finally suppose $\mathcal{C}$ itself is a small category which is a Cartesian closed category. In its presheaf category, the same construction, applied to ${\mathcal{C}}^{\mathrm{op}}$, givesThe first isomorphism uses the product's universal property: a map into $C\times A$ is precisely a pair of maps into $C$ and $A$. The next is the Yoneda lemma, and the last is the exponential adjunction in $\mathcal{C}$. Each step is natural in $C$, $A$, and $B$. Under these identifications, evaluation corresponds to $Y$ of the evaluation $B^A\times A\to B$. Hence the Yoneda embedding preserves exponentials, with the natural identification $\boxed{Y(B^A)\cong (YB{)}^{YA}}$.

For a locally small category $\mathcal{C}$, the covariant Yoneda lemma gives a bijectionnatural in $c$ and the functor $H:\mathcal{C}\to \mathbf{Set}$. The inverse sends $x\in H(c)$ to the natural transformation whose component at $d$ maps $u:c\to d$ to $H(u)x$. Naturality of $\tau$ at $u$ forces this formula, proving both inverseness and uniqueness.

Write $\mathcal{E}=\int F$ for the category of elements of $F$. Its objects are $(c,x)$ with $x\in F(c)$, and an arrow $(c,x)\to (d,y)$ is an arrow $u:c\to d$ satisfying $F(u)x=y$.

Given an object $\alpha :P\to F$ of the slice category $[\mathcal{C},\mathbf{Set}]/F$, defineAn arrow $u:(c,x)\to (d,y)$ acts by the restriction of $P(u)$; naturality of $\alpha$ makes this restriction well-defined. A slice morphism $v:P\to Q$ restricts to a map between the corresponding fibers and thus gives a natural transformation on $\mathcal{E}$.

Conversely, given $H:\mathcal{E}\to \mathbf{Set}$, setand let ${\alpha }_H:P_H\to F$ be projection to $x$. The functor laws and naturality follow from those for $F$ and $H$. On morphisms use the maps on each summand. Taking a fiber of this disjoint union recovers $H(c,x)$ canonically. Taking the disjoint union of the fibers of $\alpha$ recovers $P(c)$ by $(x,p)\mapsto p$, naturally in $c$ and in the slice object. These are inverse natural isomorphisms, provingThis is the slice of a set-valued functor category construction.

Now assume $\mathcal{C}$ is a small category, so $\mathcal{E}$ is small. Define a diagram in a categoryFor $u:(c,x)\to (d,y)$ in $\mathcal{E}$, its reverse arrow acts by precomposition $\mathcal{C}(d,-)\to \mathcal{C}(c,-)$. There is a cocone to $F$ with component $g\mapsto F(g)x$. To verify its colimit property directly, use pointwise colimits of sets. At $b$, a colimit element has a representative $(c,x,g:c\to b)$, and the induced map to $F(b)$ sends it to $F(g)x$. This map is surjective, since $z\in F(b)$ is represented by $(b,z,1_b)$. Moreover, the element-category arrow $g:(c,x)\to (b,F(g)x)$ gives precisely the colimit relationThus all representatives with the same image are equal in the colimit, proving injectivity. The resulting bijections are natural in $b$. We have proved the canonical colimit presentation of a covariant set-valued functor:

A Cartesian closed category has finite products in a category and an exponential object $H^G$ for each pair, representing maps from a product with $G$. In $[\mathcal{C},\mathbf{Set}]$, the terminal functor and binary products are computed pointwise. Define the exponential of covariant set-valued functors bySmallness makes this a set. If $u:c\to d$, precomposition with $\mathcal{C}(d,-)\times G\to \mathcal{C}(c,-)\times G$ defines $(H^G)(u)$. Functoriality is immediate from associativity of composition. Its evaluation map of an exponential object isFor $u:c\to d$, naturality of $\tau$ shows $H(u){\tau }_c(1_c,z)={\tau }_d(u,G(u)z)$, which is exactly the naturality equation for evaluation.

Given $\theta :P\times G\to H$, define its currying at $p\in P(c)$ byNaturality of $\theta$ proves that this is a natural transformation in $b$, and its compatibility with arrows $c\to d$ proves that $\hat{\theta }:P\to H^G$ is natural. Evaluation recovers $\theta$ by setting $f=1_c$. Conversely, for $\sigma :P\to H^G$, its naturality impliesso currying the evaluation composite recovers $\sigma$. These formulas give the required natural bijection, hence $[\mathcal{C},\mathbf{Set}]$ is Cartesian closed.

Finally, for every $F$ the small category $\mathcal{E}=\int F$ has a Cartesian closed functor category by the same construction. The slice equivalence above transports its terminal object, products and exponentials to $[\mathcal{C},\mathbf{Set}]/F$. The original functor category also has pointwise finite limits. Therefore it is locally Cartesian closed. Every construction here is explicit; no adjoint functor theorem is used.