Lemma 15.1.8. Let \(p\colon E \to C\) be a functor, and consider a commutative square \begin {equation*} 
Proof. Consider for any object \(Z \in E\) the following two diagrams:
The right-hand squares of both diagrams are pullback squares: for the first one since \(l\) is \(p\)-cartesian, for the second one since the image of the given square in \(C\) is a pullback square. As the functor \(p\) preserves composition, the outer rectangles of both diagrams agree, so by the pasting law the left-hand square in the first diagram is a pullback square if and only if the left-hand square in the second diagram is a pullback square. Since the former holds for all \(Z\) if and only if the given diagram is a pullback square and the latter holds for all \(Z\) if and only if \(l'\) is \(p\)-cartesian, this finishes the proof. □
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