Let $\mathsf{D}^b_h(X)$ be the full subcategory of $\mathsf{D}^b(D_X\textsf{-Mod})$ (the bounded derived category of left $D_X$-modules) consisting of the complexes with holonomic cohomology. The natural tensor product of $D_X$-modules seems to be $M\otimes_{\mathscr{O}_X}N$ with the $D_X$-action given by the Leibniz rule. But this tensor product is such that $f^!$ becomes monoidal, instead of $f^*$ as in most six-functor formalisms.
I wonder if there are natural tensor products and inner homs in $\mathsf{D}^b_h(X)$ satisfying
- The tensor-hom adjunction
- $\mathsf{D}^b_h(X)$ is symmetric monoidal
- $f^*$ is strong symmetric monoidal (that is, $f^*(M\otimes N)\cong f^*M\otimes f^*N$ and $f^* 1\cong 1$)
- The projection formula $M\otimes f_!N\cong f_!(f^*M\otimes N)$.