This is basically a question of functoriality for base change of CM morphisms.

Suppose that $f : X \to V$ is an equidimensional (dimension $d$) finite type (reduced, if it helps) Cohen-Macaulay morphism (flat with Cohen-Macaulay fibers). I'm also happy to assume that $V$ is integral, excellent and has a dualizing complex. Additionally suppose that we have $f' : X' \to V$ another equidimensional (dimension $d$) finite-type (reduced) Cohen-Macaulay morphism that factors through $f$ as below.

$$f' : X' \xrightarrow{\phi} X \to V.$$

Further suppose that $\phi$ is finite (although the question could be asked more generally for proper $\phi$, I'll phrase it for finite $\phi$). If it helps at any point, please feel free to assume that $f$ and $f'$ are Gorenstein morphisms.

Recall that $\omega_{f}[d] = f^! \mathcal{O}_V$ and that by Brian Conrad's book LINK: Google books we know that both $\omega_f$ and $\omega_{f'}$ are compatible with base change.

I'd like to conclude that the following natural map is also compatible with base change:

$$\phi_* \omega_{f'} \cong R \mathcal{H}om_{O_X}(\phi_* O_{X'}, \omega_f ) \cong \mathcal{H}om_{O_X}(\phi_* O_{X'}, \omega_f ) \to \omega_f$$.
The map can be interpreted as evaluation at 1.

In other words, I'd like to know that the trace map of $\phi$ is compatible with base change. Furthermore, it would be even good enough to prove this in the $f, f'$ Gorenstein morphisms case.

One way to do this would be as follows. If $g : T \to V$ is any other morphism and $f_T : X_T \to T$ and $f_T': X_T' \to T$ are the base changes and $g_X : X_T \to X$ is the projection, is it true that the natural map (denoted [*] below) $$g_X^* \phi_* \omega_{f'} \cong g_X^* \mathcal{H}om_{O_X}(\phi_* O_{X'}, \omega_f ) \to \mathcal{H}om_{O_{X_T}}(\phi_* O_{X_T'}, \omega_{f_T} ) \cong (\phi_T)_* \omega_{f_T'}$$ between abstractly isomorphic sheaves is an isomorphism?

Ok, I can actually prove this in a number of cases I think. But it would better to have a general reference. Here's where I can prove it (I think).

• Flat base change This is obvious for [*].

• If $\phi$ is flat Then it is finite flat, so working locally $\phi_* O_{X'}$ is locally free and things are obvious.

• Proper G1 maps I can reduce my base change to $g : T \to V$ an inclusion of a point of $V$, then the [*] map (translated back to $X_T'$) is a map between rank-1 reflexive sheaves. But a non-zero (and I can prove its non-zero) map between isomorphic rank-1 reflexive sheaves proper over a field is an isomorphism. Recall G1 means (relatively) Gorenstein in codimension 1 (which I'm actually happy to assume more generally). This last case is actually good enough for my main application, but it seems silly to assume properness.

• Regular base Again I can reduce things to the case of a $T \to V$ an inclusion of a point, and then I chase some diagrams with exts.

Ok, so at this point I am convinced it is true in general, and for my main application I need properness and G1 anyways, but it would be nice to know a reference. Or better yet, an assertion that it is obvious :-)

Does anyone know if this has appeared anywhere? This sort of functoriality of base change for $\omega_f$'s doesn't seem to appear in Brian's book (maybe I missed it), Lipman's book (although I very well could be missing something there), or in Sastry's base change paper (although maybe I am missing something again).