Proposition 2

Let E and $E'$ be coverings of B.

a) Let $f: E\rightarrow E'$ be a B-morphism. The mapping $f_b: E_b\rightarrow E'_b$ derived from $f$ is compatible with the operations of $\pi_1(B, b)$ on $E_b$ and $E'_b$ respectively. It is bijective if and only if $f$ is an isomorphism.

b) The mapping $f\mapsto f_b$ is a bijection of the set $\mathscr{C}_B(E; E')$ of B-morphisms from E into $E'$ onto the set $\mathscr{F}_{\pi_1(B,b)}(E_b; E'_b)$ of $\pi_1(B, b)$-morphisms from $E_b$ into $E'_b$.

If $f$ is a B-morphism from E into $E'$, the mapping $f_b: E_b\rightarrow E'_b$ is a $\pi_1(B, b)$-morphism (cf. III, p. 305). Moreover, two B-morphisms from E into $E'$ which coincide on the fibre $E_b$ are equal (I, p. 80, cor. 3 of prop. 6).

Let $\varphi : E_b\rightarrow E'_b$ be a morphism of $\pi_1(B, b)$-sets; let us prove that there exists a B-morphism $f$ of E into $E'$ such that $f_b=\varphi$. We may suppose that the spaces E and $E'$ are connected and non-empty, so that $E_b$ and $E'_b$ are homogeneous $\pi_1(B, b)$-sets. Let $x$ be a point of $E_b$; its stabilizer G is the image of the mapping $\pi_1(p, x)$ (III, p. 305, Theorem 1). Since the mapping $\varphi$ is a $\pi_1(B, b)$-morphism, the group G fixes the point $x'=\varphi (x)$ of $E'_b$, and is therefore contained in the image of the mapping $\pi_1(p', x')$. By Prop. 1 of III, p. 308, there exists a B-morphism $f$ of E into $E'$ such that $f(x) =x'$. The mappings $f_b$ and $\varphi$ are $\pi_1(B, b)$-morphisms of the homogeneous $\pi_1(B, b)$-set $E_b$ into $E'_b$ which agree at the point $x$; they are therefore equal.

If $f: E\rightarrow E'$ is a B-isomorphism, the mapping $f_b: E_b\rightarrow E'_b$ is bijective. Conversely, suppose that the mapping $f_b$ is bijective, and let $g: E'\rightarrow$ E be a B-morphism such that $g_b= (f_b)^{-1}$. The B-morphism $g\circ f: E\rightarrow E$ induces on $E_b$ the identical mapping; it therefore follows from the proposition that $g\circ f=$ Id$_E$. Analogously, $f\circ g=$ Id$_{E'}$, which proves that $f$ is a B-isomorphism.