Remark

Let $b$ and $b'$ be two points of B$,x$ a point of $E_b,c\in$ $\Lambda_{b,b'}(B)$ a path in B and $\widetilde{c}$ the path in E originating at $x$ which lifts $c$. For every $s\in \mathbf{I}$, let us denote by $c_s$ the path $t\mapsto c(st)$; the path in E originating at $x$ which lifts $c_s$ is the path $t\mapsto \widetilde{c}(st)$ whose term is the point $\widetilde{c}(s)$. We have therefore $\widetilde{c}(s) =x\cdot c_s$ for every $s\in \mathbf{I}$.

Let $B'$ be a topological space and let $p': E'\rightarrow B'$ be an étale and separated mapping satisfying the path lifting property. Let $f: B\rightarrow B'$ and $g: E\rightarrow E'$ be continuous mappings such that $p'\circ g=f\circ p$. For $b,b'\in B,\gamma \in \varpi_{b,b'}(B)$ and $x\in E_b$, we have:

In fact, let $c$ be a path in B whose strict homotopy class is $\gamma$, and let us denote by $\widetilde{c}$ the path in E originating at $x$ which lifts $c$; the path $g\circ \widetilde{c}$ is then the lift originating at $g(x)$ of $f\circ c$ in $E'$.

In particular, for $B = B'$ and $f=$ Id$_B$, we have

Let $p: E\rightarrow B$ and $p': E'\rightarrow B$ be étale and separated mappings satisfying the path lifting property. Let $b$ be a point of B. If $f: E\rightarrow E'$ is a B-morphism, the mapping $f_b: E_b\rightarrow E'_b$ is a $\pi_1(B, b)$-morphism.