TITLE: Does Hartshorne *really* not define things like the composition or restriction of morphisms of schemes?
QUESTION [18 upvotes]: So far as I can tell, Hartshorne's Algebraic Geometry doesn't define the composition of morphisms of schemes, or the restriction of a morphism to an open subset.  Of course it's easy enough to define these in several ways, but not having a fixed definition kind of complicates writing up rigorous solutions to some of the more foundational exercises.  Am I just missing something obvious here?

REPLY [11 votes]: It's looking like the answer is "yes."  That said, in case it helps anyone who hits this via Google, the relevant definitions are actually written down in:

EGA O, 3.5.1 - 3.5.2, 4.1.1 (composition of morphisms of ringed
spaces)
EGA O, 4.1.2 (the inclusion morphism and restriction of
morphisms)