Corollary

Let E be a Galois covering of B and let X be a covering of B, connected and nonempty. Suppose that X is a finite covering or else that the space B is locally connected. If there exists a B-morphism $h: E\rightarrow X$, the covering X is associable to the principal covering E.

In both cases, the B-morphism $h$ is a covering (I, p. 77, Cor. 3 of theorem 1, and I, p. 78, proposition 5), and a nonempty covering of a connected space is surjective (I, p. 68).