Lemma 57.15.1. Let $S' \to S$ be a morphism of schemes. The rule which sends

a smooth proper scheme $X$ over $S$ to $X' = S' \times _ S X$, and

the isomorphism class of an object $K$ of $D_{perf}(\mathcal{O}_{X \times _ S Y})$ to the isomorphism class of $L(X' \times _{S'} Y' \to X \times _ S Y)^*K$ in $D_{perf}(\mathcal{O}_{X' \times _{S'} Y'})$

is a functor from the category defined for $S$ to the category defined for $S'$.

**Proof.**
To see this suppose we have $X, Y, Z$ and $K \in D_{perf}(\mathcal{O}_{X \times _ S Y})$ and $M \in D_{perf}(\mathcal{O}_{Y \times _ S Z})$. Denote $K' \in D_{perf}(\mathcal{O}_{X' \times _{S'} Y'})$ and $M' \in D_{perf}(\mathcal{O}_{Y' \times _{S'} Z'})$ their pullbacks as in the statement of the lemma. The diagram

\[ \xymatrix{ X' \times _{S'} Y' \times _{S'} Z' \ar[r] \ar[d]_{\text{pr}'_{13}} & X \times _ S Y \times _ S Z \ar[d]^{\text{pr}_{13}} \\ X' \times _{S'} Z' \ar[r] & X \times _ S Z } \]

is cartesian and $\text{pr}_{13}$ is proper and smooth. By Derived Categories of Schemes, Lemma 36.30.4 we see that the derived pullback by the lower horizontal arrow of the composition

\[ R\text{pr}_{13, *}( L\text{pr}_{12}^*K \otimes _{\mathcal{O}_{X \times _ S Y \times _ S Z}}^\mathbf {L} L\text{pr}_{23}^*M) \]

indeed is (canonically) isomorphic to

\[ R\text{pr}'_{13, *}( L(\text{pr}'_{12})^*K' \otimes _{\mathcal{O}_{X' \times _{S'} Y' \times _{S'} Z'}}^\mathbf {L} L(\text{pr}'_{23})^*M') \]

as desired. Some details omitted.
$\square$

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