2 added 154 characters in body

Given a morphism $f:X\rightarrow Y$ between smooth complex varieties, one can define functors from the bounded derived category with holonomic cohomology on $Y$ to the same category on $X$. The easiest one is $Lf^{*}$ which can be obtained by putting a $D$-module structure on the inverse image of $\mathcal O$-modules and deriving it. From this one can get two more functors:

• $f^!:=D \circ Lf^{*} \circ D$, where $D$ is the duality functor and
• $f^{\dagger}:=Lf^{*}[dim X-dim Y]$

Now my question is, under what conditions are these two isomorphic?

Edit: These notations are bad/confusing/wrong, since they are not compatible with the formalism of six functors. See answers below for better notations.

1

# Confusion about D-modules and functors

Given a morphism $f:X\rightarrow Y$ between smooth complex varieties, one can define functors from the bounded derived category with holonomic cohomology on $Y$ to the same category on $X$. The easiest one is $Lf^{*}$ which can be obtained by putting a $D$-module structure on the inverse image of $\mathcal O$-modules and deriving it. From this one can get two more functors:

• $f^!:=D \circ Lf^{*} \circ D$, where $D$ is the duality functor and
• $f^{\dagger}:=Lf^{*}[dim X-dim Y]$

Now my question is, under what conditions are these two isomorphic?