The criterion for conic sheaves

Question

In Kashiwara and Schapira's book , chapter 3.7
It says if $X$ is a locally compact space, $F\in Ob(\mathbf{D}^+(\mathfrak{Mod}(A_X)))$, $\mu$ an action of $\mathbb{R}^+$ on $X$, consider the map $\mu,p$ from $X\times \mathbb{R}^+$ to $X$ where $p$ is the projection, $p(x,a)=x$.
Then $F$ is a conic sheaf is equivalent to $p^{-1}F\simeq \mu^{-1}F$ which is equivalent to $p^!F\simeq \mu^!F$, here $^!$ denote the dual functor of $Rf_!$.
How can we prove that these two propositions are equivalent?

Answer 1

If $X,Y$ are orientable and $f : X \to Y$ is a submersion then $f^! = f^*[d]$, proving the claim.

More generally, a submersion of manifold $f : X \to Y$ gives an isomorphism $f^! \cong f^* \otimes \omega_{X/Y}$ where $\omega_{X/Y} = f^* \underline{\Bbb{R}}_Y$ is the relative dualizing sheaf. This is a rank $1$ sheaf concentrated in degree $d$, so this is an invertible sheaf and the equivalence follows.