The Stacks project

Proof. There is a bit of abuse of notation in the lemma as strictly speaking we cannot compose $R\Gamma _ Y$ and $R\Gamma _ Z$. The meaning of the statement is simply that we are composing $R\Gamma _ Z$ with the inclusion $D_{I^\infty \text{-torsion}}(A) \to D(A)$ and then with $R\Gamma _ Y$. Then the equality $R\Gamma _ Y \circ R\Gamma _ Z = R\Gamma _{Y \cap Z}$ follows from the fact that

is right adjoint to the inclusion $D_{(I + J)^\infty \text{-torsion}}(A) \to D_{I^\infty \text{-torsion}}(A)$. Alternatively one can prove the formula using Lemma 47.9.1 and the fact that the tensor product of extended Čech complexes on $f_1, \ldots , f_ r$ and $g_1, \ldots , g_ m$ is the extended Čech complex on $f_1, \ldots , f_ n. g_1, \ldots , g_ m$. The final assertion follows from this and the cited lemma. $\square$