Corollary

Let B be a topological space, let $(E, p)$ and $(E', p')$ be two B-spaces. Suppose that the mapping $p$ is étale and separated and that the space $E'$ is locally path-connected. Then every path-continuous mapping $f: E'\rightarrow E$ such that $p\circ f=p'$ is continuous.

The mapping pr$_1: E'\times_BE\rightarrow E'$ is étale and separated (I, p. 31, prop. 8 and I, p. 27, prop. 4) and the mapping $x\mapsto (x, f(x))$ is a path-continuous section of it. By Proposition 13, it is continuous, hence $f$ is continuous.