Lemma 23.1.15. Let \(p\colon E \to C\) be a functor and let \(\phi \colon e \to e'\) be a \(p\)-cocartesian morphism. Then \(\phi \) is an isomorphism if and only if \(p(\phi )\) is an isomorphism.
Proof. Only the converse requires proof. Suppose that \(p(\phi )\) is invertible. Cocartesianness provides a morphism \(\psi \colon e'\to e\) over \(p(\phi )^{-1}\) satisfying \(\psi \phi \cong \id _e\). The morphisms \(\phi \psi \) and \(\id _{e'}\) have the same image in \(C\) and agree after precomposition with \(\phi \). The defining universal property of \(\phi \) therefore gives \(\phi \psi \cong \id _{e'}\), so \(\phi \) is invertible. □
Generated from the authoritative LaTeX source.