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.



1 Answer 1


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

  • $\begingroup$ Nice, very nice $\endgroup$
    – user16319
    Commented Jan 16, 2020 at 17:05

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