Lemma 23.1.13. The functor \(C \to *\) is both a cartesian and cocartesian fibration. Consequently, every projection functor \(\pr _D\colon C \times D \to D\) is both a cartesian and cocartesian fibration, and a morphism \((f\colon c \to c', g\colon d \to d')\) is \(\pr _D\)-cocartesian or \(\pr _D\)-cartesian if and only if \(f\) is an isomorphism.
Proof. The unique morphism in \(*\) has identity lifts. The assertion for \(\pr _D\) follows by pulling this fibration back along \(D\to *\). Using the product formula for hom animae in the defining pullback square, the cocartesian or cartesian condition reduces to requiring that precomposition or postcomposition with \(f\) induce equivalences on all hom animae of \(C\). By the Yoneda lemma, this holds if and only if \(f\) is invertible. □
Generated from the authoritative LaTeX source.