Theorem 1

Suppose that Y is path connected and that $f$ verifies the property (VK). With the preceding notations, there exists a unique group homomorphism

such that

$\mathsf{L}(v) =\delta_if_*(v)\delta^{-1}_i$ for $i\in \mathsf{I}$ and $v\in \pi_1(X,\mathsf{a}(i))$,

$\mathsf{L}(j) =\delta_{i_1(j)}\tau (j)\delta^{-1}_{i_2(j)}$ for $j\in \mathsf{J}$.

Moreover, the homomorphism $\mathsf{L}$ is surjective and its kernel is the smallest normal subgroup containing the following elements:

($R_1$)$\mathsf{r}_1(j) =j$ for $j$ in Fl($\mathsf{T}$) ;

($R_2$)$\mathsf{r}_2(j, v) =\varphi_j(v)j\psi_j(v)^{-1}j^{-1}$ for $j\in \mathsf{J}$ and $v\in \pi_1(Z,\mathsf{b}(j))$;

($R_3$)$\mathsf{r}_3(k) =\lambda_1(k)j_{12}(k)\lambda_2(k)j_{23}(k)\lambda_3(k)j_{13}(k)^{-1}$, for $k\in \mathsf{K}$.

The existence and uniqueness of the homomorphism $\mathsf{L}$ follow from the universal property of free products and free groups. Moreover, this homomorphism is the composite of the homomorphism $\varpi '(f)_{\gamma(\mathsf{a}(i_0))}$ deduced from $\varpi '(f)$ by passing to isotropy groups and of the group homomorphism $\lambda$ defined by Prop. 5 of II, p. 208, taking into account the fact that the chosen van Kampen datum determines a complete equipment of the pair $(\varpi (p_1), \varpi (p_2))$ of groupoid morphisms from $\varpi (Z)$ into $\varpi (X)$. By that proposition, the image of the homomorphism $\lambda$ is the group Coeg($f$)$_{\gamma(\mathsf{a}(i_0))}$ and its kernel is the smallest normal subgroup containing the elements defined by relations ($R_1$)$, (R_2)$ and ($R_3$). On the other hand, the groupoid morphism $\varpi '(f):$ Coeg($f$)$\rightarrow \varpi (Y)$ is an isomorphism, by definition of property (VK). The theorem is therefore proved.