The dual of the Yoneda lemma

Question

One exercise in Leinster asks to state the dual of the Yoneda lemma.

The original statement is:

Let $\mathscr A$ be a locally small category. Then $$[\mathscr A^{op},\textbf{Set}](H_A,X)\cong X(A)$$ naturally in $A\in\mathscr A$ and $X\in [\mathscr A^{op},\textbf{Set}]$.

Would the dual version be this?

Let $\mathscr A$ be a locally small category. Then $$[\mathscr A,\textbf{Set}](H^A,X)\cong X(A)$$ naturally in $A\in\mathscr A$ and $X\in [\mathscr A,\textbf{Set}]$.

(I used this answer; according to it, I just have to replace $\mathscr A$ with $\mathscr A^{op}$, which I did. I also replaced $H_A$ by $H^A$ to make sense of $[\mathscr A,\textbf{Set}](H^A,X)$.)

Answer 1

I suppose the answer that Leinster had in mind is just this:

Let $\mathscr A$ be a locally small category. Then $$[\mathscr A,\textbf{Set}](H^A,X)\cong X(A)$$ naturally in $A\in\mathscr A$ and $X\in[\mathscr A,\textbf{Set}]$.

A side note: I believe his Corollary 4.3.3 can be deduced directly from this dual version (just like Corollary 4.3.2 is deduced from his original statement of the Yoneda lemma).

Answer 2

This is sort of a mean question; there is a result which genuinely deserves to be called the (or perhaps "a") dual of the Yoneda lemma, and it isn't just given by applying the Yoneda lemma to the opposite category. But this is not at all obvious before you've seen it, and I'm not even sure it's what Leinster has in mind.

If you think of the Yoneda lemma as a "hom" statement, the dual Yoneda lemma is a "tensor" statement; you can find a discussion of it on the nLab.