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Write the Yoneda embedding as $h:\mathcal{C}\to \hat{\mathcal{C}}$. For any finite product in a category, its universal property givesThese bijections are natural, so $h$ preserves finite products. Now use the exponential in a presheaf category, followed by the Yoneda lemma and the exponential object property in $\mathcal{C}$:Every step is natural in $D$, so these component bijections form a natural isomorphismUnder this isomorphism, the image of the evaluation $B^A\times A\to B$ is the presheaf evaluation: both correspond to the same identity of the exponential under the transpose bijection. Thus the Yoneda embedding preserves exponentials, including their evaluation maps.