# Show that Yoneda embedding preserves exponentials

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

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