Monomorphisms can now be described in tensor language. He then goes on to prove that products exist. Then any product of X and Y over S is also a product over S', and vice versa. Bottom is an isomorphism, so the top is too. The rest of his construction of products is what's in Hartshorne.
Sums of products are products of sums.
Almost everything is not scheme-specific. Well, two maps into X' means two maps into X and two maps into S' . If the two maps into X' agree on Y' then they agree on Y and so the two maps into X are the same. Also, since Y' is over S', the two maps agree on S' . So there is really only one map to X and one to S' , hence one to X' .