It is well-known that for any presheaf $F \colon \mathcal{C}^{\mathrm{op}} \to \mathrm{Set}$, the category of elements (obtained by the so-called Grothendieck construction) of $F$ is a comma category $y/\ulcorner F \urcorner$ in the category $\mathrm{CAT}$ of

$$\mathcal{C} \xrightarrow{y} \widehat{\mathcal{C}} \xleftarrow{\ulcorner F \urcorner} 1.$$

**Question:** does this generalise to presheaves $F \colon \mathcal{C}^{\mathrm{op}} \to \mathrm{Gpd}$ of groupoids (or even categories)?
Mere comma categories yield discrete fibrations, hence won't give the expected answer. So the question is whether some other construction with a similar flavour could do.

Probably, if the construction does generalise, then it will also work for pseudo-functors to $\mathrm{Gpd}$ or $\mathrm{Cat}$, but I'm really interested in strict functors.

Note: I'm half-aware of another universal property of the Grothendieck construction as an oplax colimit. Is it related?