Corollary 1

Let G be a groupoid, let A be a set and let $f: A\rightarrow$ Som(G) be a mapping. If the image of $f$ meets each orbit of G, the canonical groupoid morphism $\varphi$ of $f^*G$ into G is a homotopism.

By definition of the inverse image groupoid (II, p. 166, example 4), A is the set of vertices of the groupoid $f^*G$ and one has Fl$_{a,b}(f^*G) =$ Fl$_{f(a),f(b)}(G)$ for every pair $(a, b)$ of elements of A. Moreover, one has Som($\varphi$ ) $=f$ and Fl($\varphi$ ) induces the identical mapping of Fl$_{a,b}(f^*G)$ into Fl$_{f(a),f(b)}(G)$. Consequently, the mapping Orb($\varphi$ ) is bijective and the homomorphism $\varphi_a: (f^*G)_a\rightarrow G_{f(a)}$ is an isomorphism, for every $a\in A$. The hypotheses of proposition 1 are therefore satisfied.