Natural transformations between monoid homomorphisms
For any two monoids X and Y and any two homomorphisms f and g from X to Y, a transformation from f to g is simply an element \(\alpha\) of Y. The transformation is natural iff for every \(x \in X\), we have \(g(x) \alpha = \alpha f(x)\).
I have never seen this notion referred to anywhere, so I don’t know if it’s actually useful for anything---not that I know much about the theory of monoids.
Unlike with preorders, the functor category Nat(X, Y) is not itself a monoid. Composition of natural transformations in Nat(X, Y) is just multiplication in Y.
For any three monoids X, Y, Z and any two natural transformations \(\alpha : f_1 \to g_1\) and \(\beta : f_2 \to g_2\) in Nat(X, Y) and Nat(Y, Z) respectively, we have $$ \beta \star \alpha = g_2(\alpha) \beta = \beta f_1(\alpha). $$