In category theory, is there any such thing as “compatibility” for arrow composition?

Is this a properly defined category?

1. Objects $\{P, R, S\}$
2. Arrows
• $f_{1} : P \rightarrow R$
• $f_{2} : P \rightarrow R$
• $g : R \rightarrow S$
• $h_{1} : P \rightarrow S$
• $h_{2} : P \rightarrow S$
• $id_{P} : P \rightarrow P$
• $id_{R} : R \rightarrow R$
• $id_{S} : S \rightarrow S$
3. Composition
• $g \circ f_{1} = h_{1}$
• $f_{1} \circ id_{P} = f_{1} = id_{R} \circ f_{1}$
• $f_{2} \circ id_{P} = f_{2} = id_{R} \circ f_{2}$
• $g \circ id_{R} = g = id_{S} \circ g$
• $h_{1} \circ id_{P} = h_{1} = id_{S} \circ h_{1}$
• $h_{2} \circ id_{P} = h_{2} = id_{S} \circ h_{2}$

I have 2 particular questions about it:

1. Is there any problem with omitting $h_{2}$ from composition? This seems ok because there are no arrows with domain $S$.

2. It seems like $g \circ f_{2}$ requires a composition rule, which is omitted here. But it's unclear to me whether $f_{1}$ can be thought of as "compatible" with $f_{2}$, having matching domain and codomain, such that $g \circ f_{1} = h$ would be sufficient to complete the category. Is there any such "compatibility", or does the category require a composition rule for $g \circ f_{2}$?

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You need another composition rule for $g \circ f_2$. There are two possible arrows it could be ($h_1$ or $h_2$), and the two choices of which arrow it is give rise to two different categories. More importantly, have you checked (this could be painstaking) that the compositions you have defined are associative? If not, you don't have a category.

EDIT: I just noticed that it's not hard to check that associativity holds for either assignment to $g \circ f_2$, just because there aren't very many arrows in the category to cause associativity to fail. So those are two different categories!

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Cool, thanks. In general, it hasn't been easy for me to find concrete information like this about categories. For example, given a slice category, does the slice require that the "slice arrows" be defined in the underlying category, or does it superimpose the "slice arrows" on the underlying category? All the textbooks seem to cruise over the top of these semantic distinctions, so at present my only resource is to post questions here. Do you have any suggestions for resolving this kind of thing on my own? –  Byron Hawkins Apr 24 '12 at 5:13
I think building toy examples is great. The slice arrows do have to be part of the underlying category. I learned category theory more from seeing it in action in algebraic geometry and algebraic topology than from a reference. –  user29743 Apr 24 '12 at 5:22
@ByronHawkins: Most people learn category theory with a view to applications; combinatorial/toy examples of the kind you're suggesting are unusual. Your question about slice categories makes no sense to me. A slice category is a category, not a category with added structure. –  Zhen Lin Apr 24 '12 at 8:19