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Remark 7.5.5.6 (Homotopy Invariance). Let $\operatorname{\mathcal{C}}$ be a small category and let $\alpha : \overline{\mathscr {F}} \rightarrow \overline{\mathscr {G}}$ be a natural transformation between diagrams $\overline{\mathscr {F}}, \overline{\mathscr {G}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}$. Assume that, for every object $C \in \operatorname{\mathcal{C}}$, the induced map $\alpha _{C}: \overline{\mathscr {F}}(C) \rightarrow \overline{\mathscr {G}}(C)$ is an equivalence of $\infty $-categories. Then $\alpha $ determines a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \overline{\mathscr {F}}( {\bf 0} ) \ar [r] \ar [d] & \underset {\longleftarrow }{\mathrm{holim}}( \overline{\mathscr {F}}|_{\operatorname{\mathcal{C}}} ) \ar [d] \\ \overline{\mathscr {G}}( {\bf 0} ) \ar [r] & \underset {\longleftarrow }{\mathrm{holim}}( \overline{\mathscr {G}}|_{\operatorname{\mathcal{C}}} ), } \]

where the right vertical map is an equivalence (Remark 7.5.2.7). It follows that any two of the following conditions imply the third:

$(1)$

The functor $\overline{\mathscr {F}}$ is a categorical limit diagram.

$(2)$

The functor $\overline{\mathscr {G}}$ is a categorical limit diagram.

$(3)$

The natural transformation $\alpha $ induces an equivalence of $\infty $-categories $\overline{\mathscr {F}}( {\bf 0} ) \rightarrow \overline{\mathscr {G}}( {\bf 0} )$, where ${\bf 0}$ denotes the cone point of $\operatorname{\mathcal{C}}^{\triangleleft }$.