TITLE: Show that Yoneda embedding preserves exponentials
QUESTION [5 upvotes]: Given a category $\mathbb{C}$, let $\hat{\mathbb{C}}$ be its category of presheaves.
I want to show that the yoneda embedding $y(A) = hom(-,A)$ preserves exponential  objects from $\mathbb{C}$.
I tried playing around with adjunctions, but end up in quite a mess. Hence any help or insights is appreciated.
Cheers

REPLY [7 votes]: We can string together definitions adjunctions, and the occasional use of the Yoneda lemma to get (for any object $C$ in the category $\mathbb{C}$):
$$
y(B)^{y(A)}(C) =
\operatorname{Hom}(y(C), y(B)^{y(A)}) \cong
\operatorname{Hom}(y(C) \times y(A), y(B)) \cong\\
\operatorname{Hom}(y(C \times A), y(B)) \cong
y(B)(C \times A) =
\operatorname{Hom}(C \times A, B) \cong
\operatorname{Hom}(C, B^A) =
y(B^A)(C).
$$