Let $X,Y$ be algebraic spaces, and let $X\longrightarrow Y$ be a representable morphism.
Let $S$ be a scheme and let there be a morphism $S\longrightarrow Y$. The fibre product, $X\times_Y S$, is said to be representable by a scheme. I am trying to understand in which sense this is true.
My issue is that in order to convince myself that the fibre product is representable by a scheme, I need to convince myself that each fibre thereof doesn't contain any non-trivial automorphisms (obvious since it is clearly representable by an algebraic space) and that the same fibre doesn't have any non-trivial morphisms besides identities, which is unclear to me why this should be the case, since the non-trivial morphisms of $X$ descend to give morphisms of the fibred product, do they not?