From wikipedia:

morphism: *structure preserving map from one mathematical structure to another

structure is an aggregate thing -- about the domain not about any one object in the domain.

In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions

So, in set-theory functions are structure preserving maps from a set to another.

In linear algebra, linear transformations are structure preserving maps from ...

In group theory, group homomorphisms are structure preserving maps from elements of the group (?) to ...

In Topology, continuous functions are structure preserving maps from one /region/ (?) to another.

In category theory, morphism is a broadly similar idea, but somewhat more abstract: the mathematical objects involved need not be sets, and the relationship between them may be something more general than a map.

That clears that. Arrow A->B and arrow B->C are not necessarily the same arrow (function).

Okay, so what are we doing here? Category is a set (?) of objects with arbitrary functions between them. The only property that this needs to satisfy is one of transitivity.