Remark

If, in proposition 3, one weakens the hypothesis on the mapping $f$ by supposing it only universally strict and surjective, and if $p'$ is a covering, the mapping $p$ is etale (I, p. 31, prop. 8) and separated (I, p. 27, prop. 4), but E is not necessarily a covering of B. One can find, for example, a topological space B and a locally finite closed covering $(A_i)_{i\in I}$ of B, a B-space $(E, p)$ which is not a covering but such that, for every $i\in I$, the induced $A_i$-space $E_{A_i}$ is a covering of $A_i($I, p. 146, exerc. 5). For a sufficient condition, see however corollary 4 of I, p. 77.