Edit. I am realising that I was asking the wrong question. I was (without knowing it) interested in stack comparison lemmata similar to the lemma $$Sh(Sh(C,J),can) = Sh(C,J)$$ for sheaves. So my question changes to: Which morphisms of sites induce 2-equivalences of the associated 2-categories of stacks? (my stacks are not necessarily fibered in groupoids)
Old Question.
Say I have a base category $\mathscr S = Sh(\mathbf C,J)$ of sheaves on a small site and I want to look at fibrations above $\mathscr S$. Assume $\mathbb F$ is a fibration above $\mathscr S$. What are the relations between the following conditions:
The fibration $\mathbb F$ is a stack with respect to the regular topology on $\mathscr S$, whose covers are generated by single regular epimorphisms $u: V \twoheadrightarrow U$.
The fibration $\mathbb F$ is a stack with respect to the coherent topology on $\mathscr S$. The covers in the coherent topology are generated by finite families $\{u_i: U_i \to U\}$ such that the images of the $u_i$ jointly cover $U$.
The fibration $\mathbb F$ satisfies effective descent with respect to the coproduct inclusions of finite coproducts. I believe that this is the (finitary) extensive topology.
The fibration $\mathbb F$ satisfies effective descent with respect to the coproduct inclusions of arbitrary coproducts.
The restriction of $\mathbb F$ to the small category $\mathbf C$ is a stack with respect to the topology $J$.
The fibration $\mathbb F$ is a stack with respect to the canonical topology on $\mathscr S$. The covers of the canonical topology are generated by small jointly regular epimorphic families of morphisms.
Is it for example correct that 1. and 4. together are equivalent to 6.?