Lemma 23.3.6. Let \(q\colon E \to D\) and \(p\colon D \to C\) be functors and assume that \(p\) is a left fibration. Then \(q\) is a left fibration if and only if \(pq\) is a left fibration.
Proof. As a consequence of Lemma 23.3.5, we see that a morphism \(\phi \colon e \to e'\) in \(E\) is \(q\)-cocartesian if and only if it is \(pq\)-cocartesian: in the commutative diagram
the bottom square is a pullback square, hence the top square is a pullback square if and only if the bottom one is. It follows that every morphism in \(E\) is \(q\)-cocartesian if and only if every morphism is \(pq\)-cocartesian. The claim follows. □
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