Lemma 86.7.2. Suppose we have a diagram (86.4.0.1). Let $K \in D_\mathit{QCoh}(\mathcal{O}_ Y)$. The diagram
\[ \xymatrix{ L(g')^*(Lf^*K \otimes ^\mathbf {L}_{\mathcal{O}_ X} a(\mathcal{O}_ Y)) \ar[r] \ar[d] & L(g')^*a(K) \ar[d] \\ L(f')^*Lg^*K \otimes _{\mathcal{O}_{X'}}^\mathbf {L} a'(\mathcal{O}_{Y'}) \ar[r] & a'(Lg^*K) } \]
commutes where the horizontal arrows are the maps (86.7.0.1) for $K$ and $Lg^*K$ and the vertical maps are constructed using Cohomology on Sites, Remark 21.19.3 and (86.4.1.1).
Proof.
In this proof we will write $f_*$ for $Rf_*$ and $f^*$ for $Lf^*$, etc, and we will write $\otimes $ for $\otimes ^\mathbf {L}_{\mathcal{O}_ X}$, etc. Let us write (86.7.0.1) as the composition
\begin{align*} f^*K \otimes a(\mathcal{O}_ Y) & \to a(f_*(f^*K \otimes a(\mathcal{O}_ Y))) \\ & \leftarrow a(K \otimes f_*a(\mathcal{O}_ K)) \\ & \to a(K \otimes \mathcal{O}_ Y) \\ & \to a(K) \end{align*}
Here the first arrow is the unit $\eta _ f$, the second arrow is $a$ applied to Cohomology on Sites, Equation (21.50.0.1) which is an isomorphism by Derived Categories of Spaces, Lemma 75.20.1, the third arrow is $a$ applied to $\text{id}_ K \otimes \text{Tr}_ f$, and the fourth arrow is $a$ applied to the isomorphism $K \otimes \mathcal{O}_ Y = K$. The proof of the lemma consists in showing that each of these maps gives rise to a commutative square as in the statement of the lemma. For $\eta _ f$ and $\text{Tr}_ f$ this is Lemmas 86.6.2 and 86.6.1. For the arrow using Cohomology on Sites, Equation (21.50.0.1) this is Cohomology on Sites, Remark 21.50.2. For the multiplication map it is clear. This finishes the proof.
$\square$
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