bivariate Yoneda lemma

Show that $\mathsf{Nat}(\mathsf{Hom}(-_1,X)\times\mathsf{Hom}(A,-_2),P)\cong P(A,X)$ where $P:\mathcal C^{op}\times\mathcal C\to\mathbf{Set}$. You can prove this by currying and applying each variation of the Yoneda lemma. Or, you can realize that the two variants of the Yoneda lemma are the same statement, and the above is also just a special case of the Yoneda lemma.

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Category Theory

Yoneda Lemma

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