Proposition 5

Let $G'$ be a groupoid, let $\eta ,\eta '$ be groupoid morphisms of Coh($\varphi , \psi$ ) into $G'$, and let $k$ be a mapping of Som(G) into Fl(G$'$). In order that $k$ be a homotopy joining $\eta$ to $\eta '$, it is necessary and sufficient that the following two conditions be satisfied:

(i) The mapping $k$ is a homotopy joining $\eta \circ \alpha$ to $\eta '\circ \alpha$;

(ii) For every vertex $a$ of H, one has

By definition, in order that $k$ be a homotopy joining $\eta$ to $\eta '$, it is necessary and sufficient that the following two conditions be fulfilled (recall that Som(G) = Som(Coh($\varphi , \psi$ ))) :

a) For every vertex $x$ of Coh($\varphi , \psi$ )$,k(x)$ joins $\eta (x)$ to $\eta '(x)$ ;

b) For every pair $(x, y)$ of vertices of Coh($\varphi , \psi$ ) and every arrow $f\in$ Fl$_{x,y}$(Coh($\varphi , \psi$ )), one has $\eta (f)k(y) =k(x)\eta '(f)$.

By Prop. 2, it is enough to verify condition b) when $f$ belongs to the image of Fl($\alpha$ ) or to that of $h$, so that b) is equivalent to the conjunction of the two conditions c) and d) below:

c) For $x\in$ Som(G), $y\in$ Som(G) and $g\in$ Fl$_{x,y}(G)$, one has $\eta (\alpha (g))k(y) =k(x)\eta '(\alpha (g))$;

d) For every $a\in$ Som(H), one has $\eta (h(a))k(\psi (a)) =k(\varphi (a))\eta '(h(a))$. Condition (i) is equivalent to the conjunction of a) and c), and condition (ii) is condition d), whence the corollary.