Remark 1

Let G, $G'$ be groupoids, H, $H'$ quivers, let $u: H'\rightarrow H$ be a morphism of quivers and $v: G\rightarrow G'$ a morphism of groupoids. If morphisms of quivers $\varphi ,\varphi '$ of H into G are homotopic, the morphisms of quivers $v\circ \varphi \circ u$ and $v\circ \varphi '\circ u$ of $H'$ into $G'$ are homotopic. More precisely, if $h$ is a homotopy joining $\varphi$ to $\varphi '$, the map Fl($v$)$\circ h\circ$ Som($u$) is a homotopy joining $v\circ \varphi \circ u$ to $v\circ \varphi '\circ u$.