Exercise 3, § 3

Let C be a category; we note $m$ its law of composition. Let $C^{\circ}$ be the opposite oriented graph endowed with the law of composition given by $m^{\circ}(f, g) =m(g, f)$ for every pair $(f, g)$ of composable arrows of C.

a) Prove that $C^{\circ}$ is a category. It is called the opposite category of C.

b) If C is a groupoid, prove that $C^{\circ}$ is an isomorphic groupoid to C.