Warning 7.5.3.14 (03B3)

Warning 7.5.3.14. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a strictly commutative diagram of $\infty $-categories and let $\alpha : \mathscr {F} \hookrightarrow \mathscr {F}^{+}$ denote the isofibrant replacement of Construction 7.5.3.3, and let $\theta : \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} ) \xrightarrow {\sim } \varprojlim ( \mathscr {F}^{+} )$ be the isomorphism of Proposition 7.5.3.7. We then have a diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \varprojlim ( \mathscr {F} ) \ar [r]^-{ \varprojlim ( \alpha ) } \ar [d] & \varprojlim ( \mathscr {F}^{+} ) \ar [d] \\ \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} ) \ar [r]_{ \underset {\longleftarrow }{\mathrm{holim}}(\alpha ) } \ar [ur]^{\theta }_{\sim } & \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F}^{+} ), } \]

where the outer square and the upper left triangle are commutative (Remark 7.5.3.11). Beware that the lower right triangle is usually not commutative. That is, $ \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} )$ and $\varprojlim ( \mathscr {F}^{+} )$ are isomorphic when viewed as abstract simplicial sets, but do not coincide when identified with simplicial subsets of $ \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F}^{+} )$.