Let $X$ be a smooth scheme over a field $k$ and let $\mathsf{D}_{\text{qc}}(\mathcal{D}_X)$ be the full subcategory of $\mathsf{D}(\mathcal{D}_X\mathsf{-Mod})$ composed of the complexes of $\mathcal{D}_X$-modules with quasi-coherent cohomology.

If $f:X\to Y$ is a morphism between such schemes, we have natural functors $$f_*:\mathsf{D}_{\text{qc}}(\mathcal{D}_X)\leftrightarrows \mathsf{D}_{\text{qc}}(\mathcal{D}_Y):f^!.$$

If we restrict to the subcategory of complexes with *holonomic* cohomology, then the Verdier duality functor allows us to find left adjoints $f^*$ and $f_!$ of $f_*$ and $f^!$, respectively.

**I wonder if those adjoints already exist in $\mathsf{D}_{\text{qc}}(\mathcal{D}_X)$.**

Perhaps we can use Brown representability for left adjoints (as in Neeman's book about triangulated categories) or some other adjoint functor theorem...

In Neeman's paper *The Grothendieck Duality Theorem via Bousfield's Techniques*, it is proven that $f_*$ has a **right** adjoint (even though in the text it is said that it's a left adjoint). Perhaps we can use this functor to construct our left adjoints?

`\mathsf{}`

and`\textsf{}`

: $\mathsf{D}(\mathcal{D}_X\mathsf{-Mod})$ versus $\mathsf{D}(\mathcal{D}_X\textsf{-Mod}).$ The latter gives you a hyphen rather than a minus sign. A hyphen would be standard here, if I understand you correctly. $\endgroup$