Can natural transformations be made into functors in this way?

Question

Given categories $\mathcal{C}$ and $\mathcal{D}$ and functors $F,G : \mathcal{C} \rightarrow \mathcal{D}$, can we somehow turn a natural transformation $$\nu : F \Rightarrow G : \mathcal{C} \rightarrow \mathcal{D}$$ into just another functor $$N : 2.\mathcal{C} \rightarrow \mathcal{D}?$$

I'm using $2.\mathcal{C}$ loosely as an informal notation intended to denote a category obtained by taking the coproduct $\mathcal{C} \uplus \mathcal{C}$ and then adjoining some more arrows.

http://mathoverflow.net/questions/64365/ – Martin Brandenburg Dec 10 '13 at 16:06 — Martin Brandenburg, Dec 10 '13 at 16:06

score 9 · Accepted Answer · answered Dec 10 '13 at 08:06

Yes. A natural transformation between functors $\mathcal{C} \to \mathcal{D}$ can be regarded as a functor $\mathbb{2} \times \mathcal{C} \to \mathcal{D}$, or as a functor $\mathcal{C} \to [\mathbb{2}, \mathcal{D}]$. Thus a natural transformation is like a homotopy.

Can natural transformations be made into functors in this way?

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