  The aim of this paper is to compute the Frobenius structures of some
cohomological operators of arithmetic $\ms{D}$-modules. To do this, we
calculate explicitly an isomorphism between canonical sheaves defined
abstractly. Using this calculation, we establish the relative Poincar\'{e}
duality in the style of SGA4. As another application, we compare the
push-forward as arithmetic $\ms{D}$-modules and the rigid cohomologies taking
Frobenius into account. These theorems will lead us to an analog of "Weil II"
and a product formula for $p$-adic epsilon factors.
