Yoneda embedding preserves exponentials ($Y(B^A)\cong (YB{)}^{YA}$)

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.