Remark 75.20.6. With notation as in Lemma 75.20.5. The diagram
\[ \xymatrix{ R\mathop{\mathrm{Hom}}\nolimits _ X(M, Rg'_*L) \otimes _ R^\mathbf {L} R' \ar[r] \ar[d]_\mu & R\mathop{\mathrm{Hom}}\nolimits _{X'}(L(g')^*M, L(g')^*Rg'_*L) \ar[d]^ a \\ R\mathop{\mathrm{Hom}}\nolimits _ X(M, R(g')_*L) \ar@{=}[r] & R\mathop{\mathrm{Hom}}\nolimits _{X'}(L(g')^*M, L) } \]
is commutative where the top horizontal arrow is the map from the lemma, $\mu $ is the multiplication map, and $a$ comes from the adjunction map $L(g')^*Rg'_*L \to L$. The multiplication map is the adjunction map $K' \otimes _ R^\mathbf {L} R' \to K'$ for any $K' \in D(R')$.
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