$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark (Homotopy Invariance). Let $\operatorname{\mathcal{C}}$ be a category and let $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ be a natural transformation between functors $\mathscr {F}, \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$. Then $\alpha $ determines a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \ar [dr]^{U} \ar [rr]^{T} & & \operatorname{N}_{\bullet }^{\mathscr {G}}(\operatorname{\mathcal{C}}) \ar [dl]_{V} \\ & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}). & } \]

The functor $T$ carries $U$-cocartesian morphisms of $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ to $V$-cocartesian morphisms of $\operatorname{N}_{\bullet }^{\mathscr {G}}(\operatorname{\mathcal{C}})$ (see Corollary, and therefore induces a functor of $\infty $-categories $ \underset {\longleftarrow }{\mathrm{holim}}(\alpha ): \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {G} )$. If $\alpha $ is a levelwise categorical equivalence, then $T$ is an equivalence of cocartesian fibrations over $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (Corollary, so $ \underset {\longleftarrow }{\mathrm{holim}}(\alpha )$ is an equivalence of $\infty $-categories.