Remark

If $(E, p)$ is a covering of B and if $(E', f)$ is a covering of E, the mapping $p\circ f$ is étale (I, p. 29, prop. 6, a)) and separated (I, p. 25, prop. 2, a)). But it can happen, see exercise 5, b) of I, p. 146, that the space $E'$, endowed with the mapping $p\circ f$, is not a covering of B. See however IV, p. 342, prop. 3.