of the ordinary Yoneda embedding of $C$ with the restriction functor $i^\ast$ along $i$.

When $C$ is cocomplete, the restricted Yoneda embedding has a left adjoint: the realization.

One important example of a restricted Yoneda embedding is that of the fully faithful inclusion $i : \Delta \hookrightarrow Cat$, where $\Delta$ is the simplex category. This is known as the nerve functor.