Let $f:X\to Y$ be a morphism between smooth algebraic varieties over $\mathbb{C}$.
We have natural functors $f^!:\mathsf{D}_{\text{qc}}(\mathcal{D}_Y)\to \mathsf{D}_{\text{qc}}(\mathcal{D}_X)$, $f_*:\mathsf{D}_{\text{qc}}(\mathcal{D}_X)\to \mathsf{D}_{\text{qc}}(\mathcal{D}_Y)$, and $\mathbb{D}_X:\mathsf{D}_{\text{qc}}(\mathcal{D}_X)\to \mathsf{D}_{\text{qc}}(\mathcal{D}_X)^\text{op}$ between the (unbounded, but feel free to think about the bounded case if you prefer) derived categories of left D-modules (i.e. the objects are complexes of D-modules with quasi-coherent cohomology). Naturally, we can form the diagrams below.
(The vertical arrows are simply the functors which forgets the D-module structure, and the arrows below are the functors in Grothendieck duality.)
My main question is: do these diagrams commute? If the third diagram does not commute, does it commute when restricted to $\mathsf{D}^b_{\text{coh}}$? Also, what happens for right D-modules?