While browsing some old questions I came across the following: tensor product of sheaves commutes with inverse image

It seemed like something interesting was going on in the answer, but I don't quite understand it. I understand the calculation on Hom sets, but I don't understand what is meant by:

"So $f^*\mathcal{M} \otimes_{\mathcal{O}_X} f^*\mathcal{N}$ and $f^*(\mathcal{M} \otimes_{\mathcal{O}_Y} \mathcal{N})$ represent the same functor, whence they are canonically isomorphic."

Can someone give a general statement of the fact being used here? I think this might have something to do with the Yoneda lemma, but unfortunately, I have never really been able to understand the Yoneda lemma very well. At some point I verified that the Yoneda lemma was indeed true, but my understanding was poor so it didn't really stick.