Lemma 21.38.8. Notation and assumptions as in Example 21.38.3. If there exists a cosimplicial object $U'_\bullet $ of $\mathcal{C}'$ such that Lemma 21.38.7 applies to both $U'_\bullet $ in $\mathcal{C}'$ and $u(U'_\bullet )$ in $\mathcal{C}$, then we have $L\pi '_! \circ g^{-1} = L\pi _!$ as functors $D(\mathcal{C}) \to D(\textit{Ab})$, resp. $D(\mathcal{C}, \underline{B}) \to D(B)$.

**Proof.**
Follows immediately from Lemma 21.38.7 and the fact that $g^{-1}$ is given by precomposing with $u$.
$\square$

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