Lemma 21.21.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let

be a cartesian diagram of $\mathcal{C}$. Then we have $j_{Y'/Y}^* \circ Rj_{X/Y, *} = Rj_{X'/Y', *} \circ j_{X'/X}^*$ as functors $D(\mathcal{O}_ X) \to D(\mathcal{O}_{Y'})$.

Lemma 21.21.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let

\[ \xymatrix{ X' \ar[d] \ar[r] & X \ar[d] \\ Y' \ar[r] & Y } \]

be a cartesian diagram of $\mathcal{C}$. Then we have $j_{Y'/Y}^* \circ Rj_{X/Y, *} = Rj_{X'/Y', *} \circ j_{X'/X}^*$ as functors $D(\mathcal{O}_ X) \to D(\mathcal{O}_{Y'})$.

**Proof.**
Since $j_{Y'/Y}^{-1}\mathcal{O}_ Y = \mathcal{O}_{Y'}$ we have $j_{Y'/Y}^* = Lj_{Y'/Y}^* = j_{Y'/Y}^{-1}$. Similarly we have $j_{X'/X}^* = Lj_{X'/X}^* = j_{X'/X}^{-1}$. Thus by Lemma 21.20.7 it suffices to prove the result on derived categories of abelian sheaves which we did in Lemma 21.21.1.
$\square$

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