All representable functors are continuous. This makes it possible to associate additional natural operations with them, which are absent for arbitrary presheaves.

1. What are the sufficient and what are the necessary conditions on the category $I$ for the category of continuous presheaves on $I$ to be a Grothendieck topos?
2. What are the sufficient and what are the necessary conditions on the subcanonical site I for all sheaves to be continuous? Is there a canonical way to introduce the structure of such a site on an arbitrary small category? (by canonical I mean functorial with respect to category equivalences)