I am confused about a characterisation Olsson gives for smoothness of a morphism of schemes in his book Algebraic Spaces and Stacks. This is the Theorem, on page 15 of the book,
In particular, I am confused about the phrase "for every commutative diagram of schemes". If I am understanding the proof correctly, and understanding smoothness, this result requires that $Y'$ be affine. Is this just an omission? I can't see how this can be equivalent to the definition given for smoothness otherwise.
As an aside question, when authors use the term "let $Y$ be an affine $S$-scheme", do they mean that $Y$ is an $S$-scheme and $Y$ is affine, or do they mean that $Y$ is relatively affine to $S$?