Exercise 11, § 3

Let $G$ be a groupoid; let $C$ be the set of pairs of composable arrows of $G$ and let $m: C\rightarrow$ Fl(G) be the composition law of $G$.

a) Prove that $C$ and $m$ satisfy the hypotheses (i), (ii), (iii) of Exercise 10. Denote by Γ the groupoid provided by the construction of this exercise. b) Prove that the pair formed by the mapping $e\mapsto o(e)$ from Som(Γ) into Som(G) and by the identical mapping of the set Fl(G) onto itself is an isomorphism of groupoids of Γ onto G.

This proves that a groupoid is determined by the set of its arrows and its composition law.