Is there a specific way to see a presheaf of groupoids as a colimit of representables ? As you can understand I'm looking for a similar result to the well-known fact that presheaves of sets are colimits of representables. In my case what could be the representables ?
Best
Let $\mathcal{V}$ be a Bénabou cosmos (= complete and cocomplete symmetric monoidal closed category). Then, for any small $\mathcal{V}$-category $\underline{\mathcal{A}}$, the $\mathcal{V}$-category $[\underline{\mathcal{A}}^\mathrm{op}, \underline{\mathcal{V}}]$ is the free completion of $\underline{\mathcal{A}}$ under weighted $\mathcal{V}$-colimits. Indeed, given a $\mathcal{V}$-presheaf $P : \underline{\mathcal{A}}^\mathrm{op} \to \underline{\mathcal{V}}$, we have the following formula, $$P \cong P \star_{\underline{\mathcal{A}}} Y$$ where $Y : \underline{\mathcal{A}} \to [\underline{\mathcal{A}}^\mathrm{op}, \underline{\mathcal{V}}]$ is the Yoneda embedding and $W \star_{\underline{\mathcal{A}}} F$ denotes the $\mathcal{V}$-colimit of a diagram $F : \underline{\mathcal{A}} \to \underline{\mathcal{C}}$ weighted by $W : \underline{\mathcal{A}}^\mathrm{op} \to \underline{\mathcal{V}}$.
In particular, this applies when $\mathcal{V} = \mathbf{Grpd}$ and $\mathcal{A}$ is any ordinary small category. So every presheaf of groupoids is indeed a weighted colimit of representables.