Definition 3

Let G and H be groupoids; let $\varphi$ and $\psi$ be morphisms of groupoids from H into G. Suppose that the groupoid Coeg($\varphi , \psi$ ) is transitive. Let $(a, b, c_1, c_2,T, i_0)$ be a basic equipment of the pair $(\varphi , \psi )$.

A complementary equipment is the data of a subset Z of $\Omega (\widetilde{\Gamma})$ such that the conjugacy classes $c(\mathbf{z})$ for $\mathbf{z}\in$ Z generate the kernel of the homomorphism $\beta_{a(i_0)}$ and, for every element $\mathbf{z}= ((y_1, \varepsilon_1), . . . ,(y_n, \varepsilon_n))$ of Z, of a sequence $f(\mathbf{z}) = (f_1, . . . , f_n)$ of arrows in H such that $f_k$ joins $y_k$ to the vertex $b(j_k)$, $j_k$ denoting the orbit of $y_k$ in H.

A complete equipment of the pair $(\varphi , \psi )$ is the data of a basic equipment and of a complementary equipment.