Example 2

Let H be a discrete topological group, let $f: G\rightarrow H$ be a group morphism. Endow H with the left operation of G given by $g\cdot h=f(g)h$, for $g\in G$ and $h\in H$. Let $E'= E\times^GH$ be the associated covering; it is a principal covering of group H (I, p. 107, Example 6). Let us denote by $q: E\times H\rightarrow E\times^GH$ the canonical mapping and put $x'=q(x, e)$. We have $h_{(E',x')}=f\circ h_{(E,x)}$.

In fact, let $c$ be a loop at $b$ in B, let $\gamma$ be its class in $\pi_1(B, b)$ and let $g=h_{(E,x)}$; one has therefore $x\cdot \gamma =x\cdot g^{-1}$. Let $\widetilde{c}$ be a path with origin $x$ in E; then $t\mapsto q(\widetilde{c}(t), e)$ is a path with origin $x'$ in $E'$ which lifts the path $p\circ \widetilde{c}$ in B, so that

which proves that $h_{(E',x')}(\gamma ) =f(g)$.