I'm learning about representable functors from [Vistoli notes](http://homepage.sns.it/vistoli/descent.pdf) thanks to [Charles Siegel's answer](https://mathoverflow.net/questions/10314/every-scheme-as-a-sheaf-references/10316#10316).

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?