Is there a Grothendieck correspondence for sheaves/stacks?

Question

Given a category $\mathcal{C}$, the category of elements functor sets up an equivalence of categories $$ \mathsf{DFib}(\mathcal{C}) \cong \mathsf{PSh}(\mathcal{C}), $$ whereas the Grothendieck construction sets up a $2$-equivalence $$ \mathsf{CartFib}(\mathcal{C}) \cong \mathsf{PseudoPSh}(\mathcal{C}). $$

Question: If one puts a Grothendieck topology $\mathcal{T}$ on $\mathcal{C}$ and replaces the right sides of the above ($2$-)equivalences with $\mathsf{Shv}_{\mathcal{T}}(\mathcal{C})$ and $\mathsf{Stacks}_{\mathcal{T}}(\mathcal{C})$, what should the corresponding categories on the left be?

Answer 1

The essential image of the Grothendieck construction from (weak) functors $C\to\operatorname{Cat}$ which are sheaves with respect to a Grothendieck topology $\mathcal T$ is described in Section 8.4 of the Stacks project, with the full subcategories of sheaves of groupoids and sets (more precisely, "setoids", i.e. groupoids such that all morphism sets have at most one element) described in the following sections Section 8.5 and Section 8.6, respectively.

To be precise, these sections define stacks, but you can probably compare this definition with your favourite one using the description of the Grothendieck construction in Section 4.36.