There is an analogy between the objects of study in category theory and the objects of study in homotopy theory (from topology), as well as their interactions with each other. I don't quite understand this analogy well enough myself to do it justice or make it rigorous, but as I understand it the intuition is something like this:
The objects of a category are analogous to points in a topological space.
A morphism is analogous to a path. Just as a path is a way of getting from one point to another, a morphism is a way of getting from one object to another.
A functor is analogous to a homotopy. Homotopies deform paths into different paths with different endpoints, and functors "deform" morphisms into other morphisms with different domains and codomains.
A natural transformation is analogous to a homotopy equivalence...?