When does derived pullback commute with infinite products? 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. 

Question: When does $f^*$ commute with arbitrary products?

 A: 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.
