$\newcommand{\F}{\mathscr{F}}$ $\newcommand{\I}{\mathscr{I}}$ I am trying to understand the proof of Hartshorne's Proposition III.9.3, which states that if $f: X \to Y$ is separated and quasicompact, $\F$ is quasicoherent on $X$, $u: Y' \to Y$ a flat map and $v: X \times_Y Y' = X' \to Y'$ the base change of $u$, there is an isomorphism $$u^*R^if_*(\F) \cong R^ig_* v^* (\F).$$
It is stated in the book that this is local on $Y$ and $Y'$, so we can assume they are both affine. I am having trouble seeing why this is local in that way. Once we assume $Y$ and $Y'$ are affine there are natural maps $u^*R^if_*(\F) \to R^ig_* v^* (\F)$ which turn out to be isomorphisms. The problem is that as stated in the book, this just shows that the sheaves $u^*R^if_*(\F)$ and $R^ig_* v^* (\F)$ on $Y'$ are locally isomorphic, but this obviously does not imply that they are isomorphic on all of $Y'$. (For example, any line bundle is locally isomorphic to the structure sheaf.)
To rememedy this proof, I though first to construct a global map $u^*R^if_*(\F) \to R^ig_* v^* \F$ by extending the 'push pull formula' $u^*f_*\F \to g_*v^* \scr{F}$. That way, showing this map is an isomorphism is indeed local on $Y'$ and $Y$. I know such a map exists (with no assumptions on $u$ and $v$) as it's an exercise in Vakil's notes (18.8.B) but I'm having trouble with the construction.
Say we have an injective resolution $\I^\bullet$ of $\F$ on $X$. Then, if we denote $h^i$ the cohomology of a complex, then the right exactness of $u^*$ implies there is a morphism $u^* R^if_*(\F) = u^*h^i(f_* \I^\bullet) \to h^i(u^*f_* \I^\bullet)$ which then admits a morphism to $h^i(g_*v^* \I^\bullet)$ by the push pull morphism again.
As such, we're left with finding a morphisms $h^i(g_*v^* \I^\bullet) \to R^ig_*(v^*\I^\bullet)$ but I am stuck here. So far I haven't used any of the properties of base changes or the morphisms $f$ and $g$ so I suspect there's something in this step that uses them.
Does anyone know how to finish this construction? Otherwise, do you know of a more complete proof of this theorem? I really want to understand it since it's so important.
Thanks!
EDIT: Here's an idea to construct a map $h^i(g_*v^* \I^\bullet) \to R^ig_*(v^*\I^\bullet)$. Fix an injective resolution $v^* \I^\bullet \to J^\bullet$ so that taking $g_*$ gives a map $g_*v^* \I^\bullet \to g_* J^\bullet$, so taking homology gives $h^i(g_*v^* \I^\bullet) \to R^ig_*(v^* \mathscr{I}^\bullet)$... but we really want to find a map to $R^ig_*(v^* \F)$.
It is better to use adjunction. As $u^*$ is left adjoint to $Ru_*$, constructing a map $u^*Rf_* \to Rg_*v^*$ is equivalent to constructing a map $$ Rf_* \to Ru_*Rg_*v^* \cong Rf_*Rv_*v^*, $$ where the isomorphism follows from commutativity $f \circ v = u \circ g$. Now to construct the required map, just consider the adjunction morphism $\mathrm{id} \to Rv_*v^*$ and compose it with $Rf_*$.