How is a morphism different from a function How is a morphism (from category theory) different from a function?
Intuitive explanation + maths would be great
 A: If $G$ is a group, there is a famous example that constructs a category in which morphisms are not functions: you take it to consist of a single object called "$\bullet$" and state that the morphisms (of $\bullet$ to $\bullet$, because there is no other object available) are the elements of $G$. Check for yourself that all the properties defining a category are verified by this (admittedly strange) example.
A: I'm not really sure but here is my thought. I think that is a generalization of relations (at least that the idea stemmed from them). A relation is a subset of the cartesian product of two sets, consisting of couples of elements belonging to the two sets. A relation is more general (or generic perhaps) than a function; functions are relations satisfying the condition that the elements in the domain have at most one image element in the codomain. You could picture both relations and functions with a set of arrows pointing from an element to the other from domain to codomain (which is exactly what we are thought to do in elementary math courses).
A morphism in the categorical sense does pretty much the same (and indeed is also called an arrow), except for the fact that relations are defined on sets whereas in categories you work with classes and for the fact that a relation establishes a specific "set of arrows" whereas a morphism refers to all possibile arrows between two distinct objects.
Also, a Functor seems to me to do pretty much the same as a "standard" morphism (one not in the categorical sense).
My feeling is that, in categories, switching the names morphism and functor would have generated less confusion. Functor sounds like a parent of function, though not being the same, whereas morphism would be called the same.
