Exercise 3, § 5

Let I be a preordered set and let $(G_i)_{i\in I}$ be a family of groupoids. For every pair $(i, j)$ of elements of I such that $i\leqslant j$, let $\varphi_{ji}$ be a morphism of groupoids from $G_i$ into $G_j$. We suppose that the $\varphi_{ji}$ satisfy the following conditions (cf. E, III, p. 61) :

(i) The relations $i\leqslant j\leqslant k$ imply $\varphi_{ki}=\varphi_{kj}\circ \varphi_{ji}$;

(ii) For every $i\in I,f_{ii}$ is the identity morphism of $G_i$.

Let G be the sum groupoid of the family $(G_i)$, let H be the sum groupoid of the family $(H_{ji})_{i\leqslant j}$, where one has set $H_{ji}= G_i$ for every pair $(i, j)$. Let $\varphi$ (resp. $\psi$ ) be the morphism from H into G induced by the family $(\varphi_{ji})_{i\leqslant j}$ (resp. (Id$_{G_i}$)$_{i\leqslant j}$) and let C be the coequalizer of the pair $(\varphi , \psi )$. For $i\in I$, let $\varphi_i$ be the morphism from $G_i$ into C, composed of the canonical morphism from $G_i$ into G and the canonical morphism from G into C.

a) Prove that one has $\varphi_j\circ \varphi_{ji}=\varphi_i$ for every pair $(i, j)$ such that $i\leqslant j$. b) Prove that, for every groupoid D and every family $(g_i)$, where $g_i$ is a morphism from $G_i$ into D such that $g_j\circ \varphi_{ji}=g_i$ for every pair $(i, j)$ such that $i\leqslant j$, there exists a unique morphism of groupoids $g: C\rightarrow D$ such that $g\circ \varphi_i=g_i$ for every $i\in I$.