Corollary 3

Let

${E'}^{f'}$ E

$p'p$

${B'}^f$ B

be a cartesian square. Suppose that E is a covering of B and that the space $B'$ is connected and locally connected by paths. Let $b'$ be a point of $B'$ and $b=f(b')$. In order that $(E', p')$ be a trivializable covering of $B'$, it is necessary and sufficient that, for every point $x$ of the fibre $E_b$, the image of $\pi_1(p, x)$ contain the image of $\pi_1(f, b)$.

This follows from prop. 1 and I, p. 81, cor. 5 of prop. 6.