TITLE: Yoneda embedding target
QUESTION [6 upvotes]: I'm learning about representable functors from Vistoli notes thanks to Charles Siegel's answer.
I see that any category $\mathcal C$ can be embedded into $\text{Hom}\\,(\mathcal C^{op}, \mathcal Set)$ by means of Yoneda embedding. I wonder if there are examples where the latter category would be interesting in itself, other then for these purposes?

REPLY [2 votes]: Just thought of another one: Kripke models of intuitionistic logic are (very closely related to) presheaves on the poset of possible worlds. The interpretation of intuitionistic logic in a Kripke model coincides with the internal logic of the corresponding presheaf topos.