All Questions

Are they equivalent? That is, given a sheaf of sets $\mathscr{F}$ defined on the small etale site on $X$, is there an essentially unique way to extend it to a sheaf on the big etale site on $X$? If ...

Convention: Before I start, please note that it is not going to be sufficient to assume that the topology is subcanonical or that the site $C$ has finite limits, since the application I have in mind ...