In the definition of category, there is a morphism composition law. If A, B, C are objects, and if f is a morphism from A to B, g is a morphism from B to C, then there is a corresponding morphism from A to C called the composition of f and g.
I'm wondering this law should be classified as the property of the morphism of its own, or it belongs to the structure of the whole category.
Even though both are OK for further study, but in philosophy, which is better?
More explanation about my question:
In my understanding, the definition of category is composed of 5 parts.
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The data of objects. These objects should be considered as points, and they have no property at all.
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The data of morphisms. There is a property of a morphism: It is from which object to which object.
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The composition law of morphisms as I said above.
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For every object, there is a unit morphisms satisfy … (omitted as we all know)
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The associative law of morphisms.
Part 1 and 2 are the data of the category.
Part 4 and 5 can be considered as some constraints of the category.
But part 3 is special. It is not like the data, nor the constraint of category. What should it be?
Best Answer
As I indicated in my comment on MO, this question is well-addressed by the category-theoretic distinction between stuff, structure, and properties. A category comprises
some stuff: the objects and morphisms; for simplicity we'll also consider the domain and codomain maps to be stuff (there's a bit of arbitrariness here). So the stuff of a category is its underlying graph.
some structure: the identity morphisms and composition maps. Let's call a schmategory any graph equipped with "identity edges" and "composition maps", even if they're not unital or associative.
some properties: the associativity and unit equations. So a category is a schmategory which is unital and associative.
There's a category $Cat$ of categories. Likewise, there is a category $Schmat$ of schmategories, and a category $Gph$ of graphs. There are forgetful functors $Cat \xrightarrow U Schmat \xrightarrow V Gph$. The yoga of stuff, structure, and properties tells us that $U$ forgets only properties because it is fully faithful and that $V$ forgets at most structure because it is faithful. For this reason we may say that an object of $Cat$ is an object of $Schmat$ satisfying extra properties and that an object of $Schmat$ is an object of $Gph$ equipped with extra structure.