Aggressive geometry – "Basic change" on the map étale

Let $ f colon Y to X $ a morphism of schemas. I note by $ f _ * $ and $ f ^ * $ pushing forward and removing sheaves of modules. Let $ U subseteq X $ to be an open set (Zariski), and denote by $ i colon U to X $ l & # 39; inclusion. Also, note by
begin {equation}
f_U colon f ^ {- 1} (U) to U
end {equation}

the map induced by $ f $and denote by $ j colon f ^ {- 1} (U) to Y $ l & # 39; inclusion. I think it's pretty immediate to show that, if $ mathscr F $ is a sheaf of $ mathscr O_Y $-modules, then we have an isomorphism of "base change"
begin {equation}
i ^ * f_ * mathscr F cong {f_U} _ * j ^ * mathscr F
end {equation}

of wreaths $ mathscr O_U $-modules. After all, $ i $ and $ j are just inclusions of open sets, so LHS and RHS above are given by $ V mapsto mathscr F (f ^ {- 1} (V)) $, does not matter when $ V subseteq U $.

Now, I wonder: and if we worked with a different topology, let's say the topology spreads? Assume everything as above but $ i colon U to X $ a map spread, and interpreter $ f ^ {- 1} (U) $ as the withdrawal $ Y times_ {X} U $. Is the "base change" isomorphism above still valid? I would say "yes": if so, is there a reference for that, or even something more general?