# The Stacks Project

## Tag 08P4

Remark 18.19.6. In the situation of Lemma 18.19.2 the diagram $$\xymatrix{ \textit{Mod}(\mathcal{O}_U) \ar[r]_{j_{U!}} \ar[d]_{forget} & \textit{Mod}(\mathcal{O}_\mathcal{C}) \ar[d]^{forget} \\ \textit{Ab}(\mathcal{C}/U) \ar[r]^{j^{Ab}_{U!}} & \textit{Ab}(\mathcal{C}) }$$ commutes. This is clear from the explicit description of the functor $j_{U!}$ in the lemma.

The code snippet corresponding to this tag is a part of the file sites-modules.tex and is located in lines 2239–2253 (see updates for more information).

\begin{remark}
\label{remark-localize-shriek-equal}
In the situation of Lemma \ref{lemma-extension-by-zero}
the diagram
$$\xymatrix{ \textit{Mod}(\mathcal{O}_U) \ar[r]_{j_{U!}} \ar[d]_{forget} & \textit{Mod}(\mathcal{O}_\mathcal{C}) \ar[d]^{forget} \\ \textit{Ab}(\mathcal{C}/U) \ar[r]^{j^{Ab}_{U!}} & \textit{Ab}(\mathcal{C}) }$$
commutes. This is clear from the explicit description of the functor
$j_{U!}$ in the lemma.
\end{remark}

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