The Grothendieck-Verdier duality: $$ Rf_*\big(R\mathcal{H}\textit{om}_X^\bullet(\mathcal{E}^\bullet,f^!\mathcal{F}^\bullet)\big) \cong R\mathcal{H}\textit{om}^\bullet_Y(Rf_*\mathcal{E}^\bullet,\mathcal{G}^\bullet) $$ is known to hold for $f:X\to Y$ being a proper map of noetherian schemes.

Is there a way to get rid of this requirement on schemes to be noetherian, possibly by putting some extra conditions on the morphism $f$?

For example: let $X$ be smooth (projective in necessary) and consider the projection to the second factor $f:X\times U\to U$, where $U$ is some affine scheme. Does one have the duality in this setting?

**Edit:** After reading about the topic in Neeman's and Lipman's work, I have not managed to find an explicit construction of the right adjoint $f^\times$. In what cases is the explicit construction of $f^\times$ known? What would it be in the above example?