TITLE: In what categories does the "classical" notion of function make sense?
QUESTION [5 upvotes]: I'm new to category theory, and I often struggle to choose the right level of abstraction when working with categories. I also found that many textbooks are rather inconsistent in their conventions with regards to the terminology (eg. they often interchangeably use terms like epimorphism and surjection, etc). So I wondered what's a minimal set of requirements on a category so that it makes sense to say that morphism are functions? How about Abelian categories?

REPLY [4 votes]: Beware that even in conrete categories the terms epimorphism and surjection are not interchangeable.
For example, consider 

the category Mon where objects are monoids and morphisms their algebraic homomorphisms
and the forgetful functor making it concrete being the obvious $U: \mathrm{Mon} \to \mathrm{Set}$.

Now the embedding morphism $m: (\mathbb{Z}, \cdot, 1) \to (\mathbb{Q}, \cdot, 1)$ is a monomorphism and epimorphism as can be seen by some small, but tedious calculations. However, the associated actual function given by $U(m): \mathbb{Z} \to \mathbb{Q}$ is obviously not an epimorphism in Set. Notably, in Set the terms epimorphism and surjection really coincide — at least with AoC.