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Let $C$ be an arbitrary category, $X,Y\in Obj(C)$ and $$f:X\to Y,$$ $$g:X\to Y.$$ How to define coincidence of such morphisms?

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Are you referring to the equalizer ( ) of $f$ and $g$? Not all categories have equalizers. – Shaun Ault Jan 4 '13 at 1:22
Equality between arrows or objects of an abstract category is a primitive notion, just as composition of compatible arrows is a primitive notion. – hardmath Jan 4 '13 at 1:35
Although I don't think this question is about equalizers, I'll try to post a working link. Yay, it worked! Just use the [name](url) syntax instead of bare URL. For other methods, see this post. – user53153 Jan 4 '13 at 1:37
Sorry, here's a working link: – Shaun Ault Jan 4 '13 at 1:41
@hardmath Copy into the answer box? – user53153 Jan 6 '13 at 3:33

If by "coincidence" is meant equality of morphisms (arrows) in an abstract category, one can only reply that this is an undefined concept, a primitive of what defines abstract categories.

That is, just as an abstract group has a primitive binary operation satisfying certain "equalities" but does not further define what constitutes "equality" of group elements, an abstract category is defined in terms of objects and morphisms (arrows) between them, without proposing any definition of equality of morphisms (or objects) beyond saying certain properties stated as equalities are satisfied.

In this context "primitive" has the connotation of basic concepts in terms of which all the other concepts may be defined. For example, equality of two functors might be expressed in terms of the equal assignments of objects and morphisms.

Added: If one wishes to be "bare bones" about the primitives, objects can be dispensed with in favor of doing everything in terms of morphisms, something that Eilenberg and MacLane noted in their original 1945 exposition.

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