Lemma 2.4.4 (05XR) · ScalingStacks

[05XR]

Lemma 2.4.4. In the setting of Construction 2.4.3, if $G$ is an equivalence of $\infty$ -categories, then the map $\alpha_{G}\colon T_{\mathcal{P}}\to T_{\mathcal{Q}}$ is a natural equivalence of functors.

[05XS]

Proof. Since all the steps in the construction are invariant, we may assume without loss of generality that $G$ is the identity functor and $U_{\mathcal{P}}=U_{\mathcal{Q}}$ . In this case, the map $\alpha_{G}$ is given by applying $U_{\mathcal{Q}}$ to the composition

F_{\mathcal{Q}}\xrightarrow{F_{\mathcal{Q}}u}F_{\mathcal{Q}}U_{\mathcal{Q}}F_{\mathcal{Q}}\xrightarrow{cF_{\mathcal{Q}}}F_{\mathcal{Q}}

where $u$ and $c$ are the unit and counit of the adjunction $F_{\mathcal{Q}}\dashv U_{\mathcal{Q}}$ . This composition is homotopic to the identity by the zig-zag identities. ∎