Exercise 6, § 4

Let X be a topological space that is arcwise connected, let G be a group acting continuously on X and let $x$ be a point of X such that $g\cdot x=x$ for all $g\in G$. Denote by $p$ the canonical mapping of X into $X/G$. Let H be a groupoid and let $\varphi :\varpi (X)\rightarrow H$ be a morphism of groupoids. Suppose that for every path $c$ in X and every $g\in G$, one has $\varphi ([g\cdot c]) =\varphi ([c])$, and that for every loop $c\in \Omega_x(X),\varphi ([c]) =e_{\varphi(x)}$.

Let $c_1$ and $c_2$ be paths in X such that $p\circ c_1$ and $p\circ c_2$ have the same origin and the same end point. Prove that one has $\varphi ([c_1]) =\varphi ([c_2])$.