Lemma 23.1.14 (Right cancellation of cocartesian morphisms). Let \(p\colon E \to C\) be a functor and let \(f\colon x \to y\) and \(g\colon y \to z\) be morphisms in \(E\) such that \(f\) is \(p\)-cocartesian. Then \(g\) is \(p\)-cocartesian if and only if \(gf\) is \(p\)-cocartesian.
Proof. For every other object \(w \in E\), we consider the following commutative diagram:
Since \(f\) is \(p\)-cocartesian, the right square is a pullback square. It follows from the pasting law for pullback squares that the left square is a pullback if and only if the outer square is a pullback. This proves the claim. □
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