Seems like its somehow obvious that if (X,O) and (Y,F) are affine schemes, (f,f^#) is a morphism of schemes between them (from X to Y) and U in X is an open affine, so the restriction morphism of (f,f^#) from U (with the induced sheaf structure, of course) to Y is just the composition of (res)* with f, where res is what we call the restriction map (from the sheaf structure) from O(X) to O(U) (and (res)* is the induced map on the spectrums).
I cant really understand why is it true, especially if X is not equal to the spectrum of some ring, and just isomorphic and why it seems so obvious.