Explanation of a natural transformation in group theoretic terms

Question

In category theory, a functor looks like homomorphism between categories. Keeping that analogy in mind, can a natural transformation be described by (or restricted to) group theoretic terms? For example, can they be described by groups of homomorphisms?

Answer 1

Recall that a small category with only one object in which all morphisms are invertible is precisely a group, and functors between such categories are group homomorphisms. Thus we can understand natural transformations as a generalization of some notion of equivalence of group homomorphisms.

If we have two group homomorphisms $f,g:X\to Y$ then a natural transformation between $f$ and $g$ is an element $\eta\in Y$ such that for any $x\in X, \eta f(x)=g(x)\eta$, i.e. $f(x)=\eta^{-1}g(x)\eta$. Thus two group homomorphisms are related by a natural translation iff they differ by an inner automorphism.