Is this a properly defined category?
- Objects $\{P, R, S\}$
- Arrows
- $f_{1} : P \rightarrow R$
- $f_{2} : P \rightarrow R$
- $g : R \rightarrow S$
- $h_{1} : P \rightarrow S$
- $h_{2} : P \rightarrow S$
- $id_{P} : P \rightarrow P$
- $id_{R} : R \rightarrow R$
- $id_{S} : S \rightarrow S$
- 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:
Is there any problem with omitting $h_{2}$ from composition? This seems ok because there are no arrows with domain $S$.
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 $g \circ f_{2}$?