= Yoneda embedding preserves exponentials
{c}
{title2=$Y(B^A)\cong(YB)^{YA}$}
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 $\operatorname{Nat}(YC\times YA,YB)$. 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.
Back to article page
