Does the canonical line bundle coincide with dualizing sheaf when the base scheme is arbitrary

Question

Let $X$ be a smooth projective curve over an algebarically closed field $k$.

Consider the commutative diagram where the left square is cartesian.

where $S,S'$ are schemes over $k$ and $X_S:=X\times S$ and $X_{S'}:=X\times S'$.

Let $\omega_{X|k}$ be the canonical bundle of $X$ and let $\omega_{X_S/S}:=p^*(\omega_{X|k})$.

1. Is it true that $\omega_{X_S/S}$ is a dualizing sheaf for $X_S$?
2. Similarly, is $\omega_{X_{S'}/S'}:=g^*(\omega_{X_S/S})$ a dualizing sheaf for $X_{S'}$?

I can see that $\omega_{X_S/S}$ and $\omega_{X_{S'}/S'}$ are canonical line bundles respectively on $X_{S},X_{S'}$, but I couldn't find anything prooving that the canonical bundle coincides with the dualizing sheaf in the case where the base scheme is arbitrary like with $S,S'$.(when it is a field, Hartshorne III, cor 7.12 gives an answer in that special case).

Thank you for your help.