Minimal site that induces a stack from a psuedofunctor

Question

I'm working with Vistoli's definition of stacks [1]

Let $\Phi:C^{op}\to {Cat}$ be a psudeofunctor.

Is there always a minimal Grothendieck topology on $C$ such that it induces a stack? If not, are the conditions on $C$ or $\Phi$ that one can impose to derieve the existance of such a topology (maybe along with a description)?

1 - Notes on Grothendieck topologies, fibered categories and descent theory

Answer 1

Well, every functor is certainly a stack with respect to the minimal Grothendieck topology, namely, that in which the only coverings are isomorphisms. So it is more reasonable to ask for a maximal topology with respect to which $\Phi$ is a stack. For this, you can just take the topology consisting of sieves $S$ such that $\Phi$ is a stack with respect to every pullback of $S$. There's nothing special about stacks, here-you tell the same story for strict sheaves.