Let $G=(G_1, G_0, s, t, u, i,\circ)$ be a groupoid, where $s, t$ are source and target maps, $i$ is the inverse, $u$ is the unit, and $\circ$ is the composition.

Denote $\underline{G_1}, \underline{G_0}$ the trivial groupoid on $G_1, G_0$ respectively. There are morphisms of Lie groupoids $s:\underline{G_1}\to \underline{G_0}$ and $t: \underline{G_1}\to \underline{G_0}$, and $u:\underline{G_0}\to G$. There is a natural transformation $us\Rightarrow ut:\underline{G_1}\to G$ given by $id: G_1\to G_1$.

Is it true that the 2-commutative diagram

$$\underline{G_1}⇉^s_t \underline{G_0}\xrightarrow{u} G$$

is a 2-coequaliser diagram in the 2-category of groupoids?

If this is not the case, how to correct it? I learn from Thomason’s homotopy colimit theorem that the 2-colimit of the nerve $\Delta^{op}\to NG$ and $G$ has the same homotopy type. A relevant statement is a quotient stack is a 2-coequalizer.