I am trying to understand the proof of proposition 4.1 of Milne's book Étale Cohomology (p.120) and I am getting really confused with some points of the reverse implication: if I understand correctly his notations he claims that the isomorphisms of $S$-schemes with $G$-action $(X\times_S U_i) \cong (G \times_{S} U_i)$ that hold for all $i \in I$ induce an isomorphism of schemes with $G$-action from $X \times_S U$ to $G\times_S U$, where $U = \bigsqcup_{i\in I} U_{i}$. So it seems that pullback commutes with disjoint union here. Why is it true?
I have read in the Introduction to Étale Cohomology of Tamme, chapter 0 paragraph 3 that if $F: \mathcal{C} \rightarrow \mathcal{C'}$ is a functor between arbitrary categories that admits a left adjoint functor then this left adjoint functor commutes ith inductive limit. It could be an explanation to my problem but I still have some doubts on the following facts:
- Is the category of faithfully flat locally of finite type schemes abelian with arbitrary direct sums?
- In this case $^{ad}F$ must be the pullback of faithfully flat locally of finite type schemes, so $F$ should be the pushout of such type schemes but does it always exists?
Is there a more direct way for showing the existence of this induced isomorphism?
Many thanks for your help!