A category consists of objects and morphisms between objects. The term "morphism" is a little bit misleading (they are not required to morph anything);
so morphisms are frequently called "arrows", to stress their abstract nature.
I'll use the term "arrow" except when an arrow represent some kind of function, in which case I'll call it a "morphism". But it's still just an arrow to me.
We do not care about the nature of object and arrows; all we need are the following properties: