Straightening and unstraightening

Theorem 23.2.1 (Straightening/Unstraightening, Lurie (2009), Hebestreit et al. (2021)). Let \(C\) be a small \(\infty \)-category. Then there exist equivalences of \(\infty \)-categories \[ \Str ^{\cc }\colon \Cocart (C) \iso \Fun (C,\Cat _{\infty }) \] and \[ \Str ^{\ct }\colon \Cart (C) \iso \Fun (C\catop ,\Cat _{\infty }). \] Furthermore, these equivalences are natural¹ in \(C\), in the sense that for every functor \(F\colon C \to D\) the following two squares commute:

Finally, the composites \begin {align*} \Cat _{\infty } \simeq \Cocart (*) &\xrightarrow {\Str ^{\cc }} \Fun (*,\Cat _{\infty }) \simeq \Cat _{\infty } \\ \Cat _{\infty } \simeq \Cart (*) &\xrightarrow {\Str ^{\ct }} \Fun (*,\Cat _{\infty }) \simeq \Cat _{\infty } \end {align*}

are (isomorphic to) the identity functors.