This is similar to a question I asked recently, but this time specifically for groupoids.
Suppose $f: A \rightarrow B$ is a groupoid morphism. Let $f^\ast: [B, \text{Set}] \rightarrow [A, \text{Set}]$ be the functor given by precomposition by $f$, so that $G: B \rightarrow \text{Set}$ maps to $G \circ f : A \rightarrow \text{Set}$. I'm wondering if the following statements are true:
1) $f$ is essentially surjective and full if and only if $f^\ast$ is full an faithful.
2) $f$ is an equivalence if and only if $f^\ast$ is an equivalence.
I think I have the forward directions for both of these statements, and proofs by contradiction for the reverse direction of (1). Zhen Lin's answer to my previous question tells me that (2) holds, since groupoids are isomorphic to their Cauchy completions (their only indempotents are identities). However, I would really appreciate a proof sketch of (2), and (constructively) the reverse direction of (1), since I am pretty stumped.
Thanks.