Proposition 3

The pair (Coeg($\varphi , \psi$ )$, \gamma$ ) has the following universal property:

a) We have $\gamma \circ \varphi =\gamma \circ \psi$.

b) Let $G'$ be a groupoid and let $\theta : G\rightarrow G'$ be a groupoid morphism such that $\theta \circ \varphi =\theta \circ \psi$. There exists a unique groupoid morphism $\overline{\theta}:$ Coeg($\varphi , \psi$ )$\rightarrow G'$ such that $\overline{\theta}\circ \gamma =\theta$.

Let $a$ be a vertex of H. The arrow $h(a)$ of Coh($\varphi , \psi$ ) joins $\alpha (\varphi (a))$ to $\alpha (\psi (a))$. By definition of the groupoid Coeg($\varphi , \psi$ ), the origin and the end of the arrow $\beta (h(a))$ are equal; therefore we have $\gamma (\varphi (a)) =\gamma (\psi (a))$.

Let $f$ be an arrow of H; if $a$ denotes its origin and $b$ its end, we thus have

Taking the image by $\beta$ of the two members of this equality, one obtains the relation $\gamma (\varphi (f)) =\gamma (\psi (f))$. This proves assertion a).

Let us prove b). The mapping $\eta :$ Som(H) $\rightarrow$ Fl(G$'$) which associates with every vertex $a$ of H the arrow $e_{\theta(\varphi(a))}$ of $G'$ is a homotopy joining $\theta \circ \varphi$ to $\theta \circ \psi$. By the universal property of cohomotopizers (II, p. 185, prop. 3), there exists a unique groupoid morphism $\theta_1:$ Coh($\varphi , \psi$ )$\rightarrow G'$ such that $\theta_1\circ \alpha =\theta$ and $\theta_1(h(a)) =e_{\theta(\varphi(a))}$ for every vertex $a$ of H. By prop. 7 of II, p. 176, this latter property implies the existence of a unique groupoid morphism $\overline{\theta}:$ Coeg($\varphi , \psi$ )$\rightarrow G'$ such that $\overline{\theta}\circ \beta =\theta_1$. One then has $\overline{\theta}\circ \gamma =\theta_1\circ \alpha =\theta$.

Conversely, if $\overline{\theta}':$ Coeg($\varphi , \psi$ )$\rightarrow G'$ is a groupoid morphism such that $\overline{\theta}'\circ \gamma =\theta$, one has $(\overline{\theta}'\circ \beta )\circ \alpha = (\overline{\theta}\circ \beta )\circ \alpha$, whence $\overline{\theta}'\circ \beta =\overline{\theta}\circ \beta$ by II, p. 185, prop. 3, whence $\overline{\theta}'=\overline{\theta}$ by prop. 7 of II, p. 176. This proves the uniqueness of $\overline{\theta}$, hence assertion b).