Remark 3

Let

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

$p'p$

${B'}^f$ B

be a cartesian square. If the mapping $p$ is étale and if the mapping $f$ is strict, the mapping $f'$ is strict. In fact, under these hypotheses, every point of E possesses an open neighbourhood U such that $p$ induces a homeomorphism of U onto the open set $p(U)$. The mapping $p'$ then induces

a homeomorphism of $(\overset{-1}{f}')(U)$ onto $\overset{-1}{f}(p(U))$. The mapping $f$ induces

a strict mapping of $\overset{-1}{f}(p(U))$ onto $p(U)$ (I, p. 20) and the mapping

$f'$ therefore induces a strict mapping of $(\overset{-1}{f}')(U)$ into U. It follows that the mapping $f'$ is strict ( I, p. 23, corollary 2).