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?



Is there a Grothendieck correspondence for sheaves/stacks?
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](https://stacks.math.columbia.edu/tag/0268), 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](https://stacks.math.columbia.edu/tag/02ZH) and [Section 8.6](https://stacks.math.columbia.edu/tag/042Y), 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](https://stacks.math.columbia.edu/tag/02XU).


