Let $f:X \to Y$ be a morphism of reasonable schemes (qcqs). Let $f^\*: D(Y) \to D(X)$ be the pullback defined on the derived unbounded categories of quasi-coherent sheaves. 



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> Question: **When does $f^\*$ commute with arbitrary products?**
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When does derived pullback commute with infinite products?
The functor $f^\*: D(Y) \to D(X)$ preserves limits if $Y$ is noetherian and $f$ is proper flat with Gorenstein fibers, for example if $f$ is smooth and proper. Under these assumptions: $f^!$ is right adjoint to $f\_\*$ by properness, $f^!(\mathcal F)=f^!(\mathcal O\_Y)\otimes f^\*(\mathcal F)$ by finite tor-amplitude, and $f^!(\mathcal O\_Y)$ is invertible by Gorenstein. Hence, $f^\*$ is right adjoint to $f\_\*(f^!\mathcal O\_Y\otimes -)$. 


See <http://stacks.math.columbia.edu/tag/0C02> for a discussion of Gorenstein morphisms, particularly Lemma 45.43.8, and section 7 in <https://arxiv.org/pdf/1406.7599v1.pdf> for the relevant properties of $f^!$.


**ETA** According to pbelmans' comment above, the Gorenstein assumption is not actually needed: if $Y$ is arbitrary and $f$ is proper, pseudo-coherent, and of finite tor-amplitude, then $f^\*$ is right adjoint to $f\_\*(f^!\mathcal O\_Y\otimes -)$. Combine Corollary 6.4.2.7 in <http://www.math.harvard.edu/~lurie/papers/SAG-rootfile.pdf> and Theorem 1.7 in <https://arxiv.org/abs/1501.01999>.


