Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 7.5.3.11. In the situation of Proposition 7.5.3.7, the isomorphism $\theta : \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F} ) \xrightarrow {\sim } \varprojlim ( \mathscr {F}^{+} )$ fits into a commutative diagram

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

where $\iota $ is the comparison map of Remark 7.5.2.10 and $\alpha : \mathscr {F} \hookrightarrow \mathscr {F}^{+}$ is the natural transformation appearing in Construction 7.5.3.3.