Corollary 23.2.7. The straightening equivalences \(\Str ^{\cc }\) and \(\Str ^{\ct }\) restrict to equivalences \[ \Str ^{\cc } \colon \LFib (C) \iso \Fun (C,\An ) \qquadtext { and } \Str ^{\ct } \colon \RFib (C) \iso \Fun (C\catop ,\An ). \]
Proof. Given a cocartesian fibration \(p\colon E \to C\), the individual values of \(\Str ^{\cc }(p)\) are precisely the fibers of \(p\). In particular, \(\Str ^{\cc }(p)\) lands in the subcategory \(\An \subseteq \Cat _{\infty }\) if and only if all fibers are animae. By Proposition 23.1.16, this is equivalent to \(p\) being a left fibration, proving the first equivalence. The dual discussion provides the right equivalence. □
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