Corollary

Let B be a connected and locally connected topological space. Let E and $E'$ be coverings of B. Let $b$ be a point of B. Let $f: E'\rightarrow E$ be a B-morphism and let $f_b: E'_b\rightarrow E_b$ be the mapping deduced from $f$ by restriction to the fibres. If the mapping $f_b$ is injective ( resp. surjective, resp. bijective), the same is true of the mapping $f$.

Let us denote by $p$ the projection of the B-space E. By the proposition, $(E', f)$ is a covering of E; its image $f(E')$ is open and closed in E. Let U denote the complement of $f(E')$ in E, so that the B-space $(U, p|U)$ is a covering of B (I, p. 80, corollary 1). If the mapping $f_b$ is surjective, the fibre at $b$ of this covering is empty. Since B is a connected space, U is then empty, hence $f$ is surjective.

The set V of points of E at which the fibre of $f$ has exactly one element is open and closed. The B-spaces $(V, p|V)$ and $(f(E'), p|f(E'))$ are coverings of B (loc. cit.) and the canonical mapping $i: V\rightarrow$ $f(E')$ is a B-morphism. If the mapping $f_b$ is injective, the mapping $i_b$ is surjective, hence $i$ is surjective by what precedes, and one has $V =f(E')$. Consequently, the mapping $f$ is injective.