Remark 2

Let G be a groupoid, let H be a quiver, and let $\varphi , \psi$ be morphisms of quivers of H into G. Let us denote by $j$ the canonical morphism of H into Grp(H), and let $\overline{\varphi}$ and $\overline{\psi}$ be the morphisms of groupoids of Grp(H) into G such that $\overline{\varphi}\circ j=\varphi$ and $\overline{\psi}\circ j=\psi$.

Recall that Som(H) = Som(Grp(H)).

A homotopy $h:$ Som(H) $\rightarrow$ Fl(G) joining $\varphi$ to $\psi$ is a homotopy joining $\overline{\varphi}$ to $\overline{\psi}$.