  Let $\V$ be a mixed characteristic complete discrete valuation ring, let $\X$
and $\Y$ be two smooth formal $\V$-schemes, let $f_0$ : $X \to Y$ be a
projective morphism between their special fibers, let $T$ be a divisor of $Y$
such that $T_X := f_0 ^{-1} (T) $ is a divisor of $X$ and let $\M \in D
^\mathrm{b}_\mathrm{coh} (\D ^\dag_{\X} (\hdag T_X)_{\Q})$. We construct the
relative duality isomorphism $ f_{0T +} \circ \DD_{\X, T_X} (\M) \riso \DD_{\Y,
T} \circ f_{0T +} (\M)$. This generalizes the known case when there exists a
lifting $f : \X \to \Y$ of $f_{0}$. Moreover, when $f_0$ is a closed immersion,
we prove that this isomorphism commutes with Frobenius.
