Proposition 4.3.13. For any pointed \(\infty \)-category \(D\) with finite limits, the composite map \[ \Omega ^3X \simeq \Omega ^2(\Omega X) \xrightarrow {\cong } \Omega (\Omega ^2X) \simeq \Omega ^3X \] is homotopic to the identity map on \(\Omega ^3X\), naturally in \(X \in D\).
Proof. Precomposition with \((-)_+\colon \An ^{\fin }\to \An _*^{\fin }\) defines a functor \[ \Fun ^{\lex }((\An _*^{\fin })\catop ,D) \longrightarrow \Fun ^{\lex }((\An ^{\fin })\catop ,D). \] An inverse sends \(F\) to the functor \[ (Y,y)\longmapsto \fib (F(Y)\to F(*)). \] This is pointed and left exact. The two inverse identities follow from \(Y_+=Y\sqcup *\) and from the pushout \(Y_+\sqcup _{S^0}*\simeq (Y,y)\). The result now follows from Proposition 4.3.11. □
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