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Write the Yoneda embedding as $h:\mathcal{C}\to \hat{\mathcal{C}}$. For any finite product in a category, its universal property givesThese bijections are natural, so $h$ preserves finite products. Now use the exponential in a presheaf category, followed by the Yoneda lemma and the exponential object property in $\mathcal{C}$:Every step is natural in $D$, so these component bijections form a natural isomorphismUnder this isomorphism, the image of the evaluation $B^A\times A\to B$ is the presheaf evaluation: both correspond to the same identity of the exponential under the transpose bijection. Thus the Yoneda embedding preserves exponentials, including their evaluation maps.

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 Cartesian closed category has finite categorical products and, for every $P,Q$, an exponential object $Q^P$ with an evaluation morphism $\mathrm{ev}:Q^P\times P\to Q$ such that composition with evaluation gives natural bijectionsFinite categorical products in the presheaf category are pointwise; in particular its terminal object is the constant singleton categorical presheaf. For categorical presheaves $P,Q$ defineFor $u:d\to c$, restriction sends $\alpha$ to $\alpha \circ (H_u\times 1_P)$. Composition and identity laws follow from those of the Yoneda embedding, so this is a categorical presheaf.

Define evaluation at $c$ by ${\mathrm{ev}}_c(\alpha ,x)={\alpha }_c(1_c,x)$. For $u:d\to c$, naturality of $\alpha$ gives $Q(u){\alpha }_c(1_c,x)={\alpha }_d(u,P(u)x)$, which is the evaluation of the restricted pair. Hence evaluation is a natural transformation.

Given $t:R\times P\to Q$, its currying sends $z\in R(c)$ to the transformation whose component at $a$ isNaturality in $a$ follows from that of $t$, and naturality in $c$ follows by composing $v$ with the relevant arrow. Evaluating $\hat{t}$ at $(1_c,x)$ recovers $t_c(z,x)$. Conversely, if $s:R\to Q^P$, naturality of $s$ along $v:a\to c$ says that evaluating $s_a(R(v)z)$ at $(1_a,x)$ gives $(s_c(z){)}_a(v,x)$. Thus currying the evaluation of $s$ recovers $s$. These constructions are inverse and natural, proving the universal property. This is the exponential in a presheaf category, and proves the presheaf category is Cartesian closed.

The Yoneda embedding preserves categorical products: for any $a$, maps $a\to c\times Y$ correspond naturally to pairs of maps $a\to c$ and $a\to Y$. Hence $H_{c\times Y}\cong H_c\times H_Y$. Using the exponential in a presheaf category, the Yoneda lemma and the exponential universal property in $\mathcal{C}$ givesEvery bijection is natural in $c$, since it is given by precomposition or the natural exponential adjunction. Thus these component bijections are a natural isomorphism of categorical presheaves:

For a small Cartesian closed category, the Yoneda embedding into its presheaf category preserves exponential objects. At $C$, the exponential in a presheaf category has value $\mathrm{Nat}(YC\times YA,YB)$. Since $YC\times YA\cong Y(C\times A)$, the Yoneda lemma identifies this with $\mathcal{C}(C\times A,B)\cong \mathcal{C}(C,B^A)$. Every identification is natural in all three objects.