Proposition 4

Let X be the product space of a family $(X_j)_{j\in J}$ of topological spaces. The morphism of the groupoid $\varpi (X)$ into the product of the groupoids $\varpi (X_j)$, for $j\in J$, defined by the family of morphisms $(\varpi$(pr$_j$))$_{j\in J}$ is an isomorphism.

Let $\varphi$ denote this morphism of groupoids. The mapping deduced from it by passing to the vertices is the identity mapping $X\rightarrow \prod_jX_j$. Let $x= (x_j)$ and $y= (y_j)$ be two points of X. Let $\varphi_{x,y}$ denote the mapping of $\varpi_{x,y}(X)$ into $\prod_j\varpi_{x_j,y_j}(X_j)$ deduced from $\varphi$. If for every $j\in J,c_j:\mathbf{I}\rightarrow$ $X_j$ is a path joining $x_j$ to $y_j$, the mapping $t\mapsto (c_j(t))$ is a path in X joining $x$ to $y$ (TG, I, p. 25, prop. 1). This proves that $\varphi_{x,y}$ is surjective. Let $c$ and $d$ be two paths in X joining $x$ to $y$. Suppose that for every $j\in J$ there exists a strict homotopy $\sigma_j:\mathbf{I}\times \mathbf{I}\rightarrow X_j$ joining pr$_j\circ c$ to pr$_j\circ d$. Then the mapping $(t, s)\mapsto (\sigma_j(t, s))$ of $\mathbf{I}\times \mathbf{I}$ into X is a strict homotopy joining $c$ to $d($loc. cit.). This proves that $\varphi_{x,y}$ is injective.