I'm just beginning to learn category theory. So far, the basic examples (like Set) are making sense. But I'm having a little trouble getting my head around the fundamentals.
Suppose I try to define a category with objects A, B, and C. There's an arrow (f) from A to B, an arrow (g) from B to C, and two arrows (h1 and h2) from A to C (in addition to the identity arrows).
Now it must obey the composition axiom:
For any two arrows f : A → B, g : B → C, where src(g) = tar (f), there exists an arrow g ◦ f : A → C, ‘g following f’, which we call the composite of f with g.
It seems we have two choices for g ◦ f. Since we're not trying to give the category any meaning, it shouldn't matter which we pick. But does the choice matter in identifying the category? Does picking h1 give a "different" category than picking h2? If so, shouldn't it be part of the category's definition? If not, then is it meaningless to ask "which one is the composite?" Perhaps another way of asking this is: does composition have to mean any (one) thing?
My question probably doesn't make a lot of sense, but hopefully there are enough clues in here for someone to correct my confusion.
Edit: Perhaps restating the axiom in these different ways clarifies my confusion.
a) For any two arrows f : A → B, g : B → C, where src(g) = tar (f), there exists at least one arrow g ◦ f : A → C, ‘g following f’, which we could call the composite of f with g.
b) For any two arrows f : A → B, g : B → C, where src(g) = tar (f), there exists an arrow g ◦ f : A → C, ‘g following f’, that is the composite of f with g.