If I were the person creating category theory, I wouldn’t have been interested in left and right cancellation. I may or may not notice it, but it doesn’t feel fundamental enough for me to define it generally or give it a name. Question: if you tell me a morphism is epic or monic, what other insight do I gain about the morphism that is equivalent but conceptually what we actually use? (since just taking about cancellations seems unnatural and unworthy of a name to me right now).
As an example I’ve thought about stuff like “information is lost” but that doesn’t feel like it works somehow.
EDIT: ok I've heard they're supposed to be categorical generalisations of injections and bijections, but from the definition they don't always behave in ways you'd expect (just incase, I can find some places if you want). Is there a more general but relatable way to think about them that eliminates this occasional weirdness?
The concept of monomorphism is one of the most fundamental concepts of category theory. It is used to define posets of subobjects, which are used to define well-poweredness, which is one of the properties arising in the Special Adjoint Functor Theorem, one of the most fundamental theorems of category theory.
Monomorphisms and subobjects are deeply connected, as well as their duals epimorphisms and quotient objects, appearing almost in every topic of category theory. For example, in sheaf theory: the basic notion of a sieve can be defined as a subobject of a hom-functor, which may be presented as a monomorphism in the category of presheaves.
Monomorphisms and epimorphisms are generalizations of very important notions in different algebraic categories, what explains their wide usage in homological algebra. For example, in the theory of abelian categories, a monomorphism is the same as a kernel, which is also the very basic and fundamental notion.
So you can see that monics and epics are fundamental notions and are necessary to define in any "creation" of category theory. That's why you can find their definitions at the first pages of every category theory textbook.