In category theory, arrows between categories are functors, arrows between functors are natural transformations, so a natural question is to ask what are arrows between natural transformations ?
I've got an idea to generalize that, but I don't know if its any useful or known. Let $F$ and $G$ be two functors from categories $\mathcal{C}$ to $\mathcal{D}$, and let $\eta: F \Rightarrow G$. My idea is to see a natural transformation as a functor from $\mathcal{C}$ to the arrow category of $\mathcal{D}$.
Let $\eta : \mathcal{C} \rightarrow Arr(\mathcal{D})$ be a functor. We say that $\eta$ is a natural transformation between $F$ and $G$ if for all objects $A$, $B$, and arrows $f : A \rightarrow B$ in $\mathcal{C}$, we have the following conditions :
1) $\eta(A) = \eta_A : F(A) \rightarrow G(A)$ (we call it like that even though it is not the component of a natural transformation, it is only an object in the arrow category of $\mathcal{D}$)
2) $\eta$ sends $f$ to $\eta(f) = (F(f), G(f))$ such that $\eta_B \circ F(f) = G(f) \circ \eta_A$ (that is, the square commutes). Such an $\eta$ encodes what a natural transformation is, and it is easily seen what a "natural transformation" between two such functors is. It seems that one could easily generalize that up to infinity.
Does anyone know if such an idea is known, useful or not, or if there's better ?
edit: looks like it's now well defined