Theorem 23.3.1. Consider a cocartesian functor
where \(p\) and \(p'\) are cocartesian fibrations between small \(\infty \)-categories. Then \(F\) is an equivalence of \(\infty \)-categories if and only if for every object \(x\) of \(C\), the induced functor on fibers \(F_x\colon E_x \to E'_x\) is an equivalence.
Proof. Under the equivalence \(\Str ^{\cc }\colon \Cocart (C) \iso \Fun (C,\Cat _{\infty })\), \(F\) corresponds to a natural transformation \(\Str ^{\cc }(F)\colon \Str ^{\cc }(p) \to \Str ^{\cc }(p')\), and this natural transformation is a natural equivalence if and only if it is a pointwise equivalence. □
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