TITLE: Natural transformations as categorical homotopies 
QUESTION [59 upvotes]: Every text book I've ever read about Category Theory gives the definition of natural transformation as a collection of morphisms which make the well known diagrams commute.
There is another possible definition of natural transformation, which appears to be a categorification of homotopy:

given two functors $\mathcal F,\mathcal G \colon \mathcal C \to \mathcal D$ a natural transformation is a functor $\varphi \colon \mathcal C \times 2 \to \mathcal D$, where $2$ is the arrow category $0 \to 1$, such that $\varphi(-,0)=\mathcal F$ and $\varphi(-,1)=\mathcal G$.

My question is:

why doesn't anybody use this definition of natural transformation which seems to be more "natural" (at least for me)?  

(Edit:) It seems that many people use this definition of natural transformation. This arises the following question: 

Is there any introductory textbook (or lecture) on category theory that introduces natural transformation in this "homotopical" way rather then the classical one? 

(Edit2:) Some days ago I've read a post in nlab about $k$-transfor. In particular I have been interested by the discussion in the said post, because it seems to prove that the homotopical definition of natural transformation should be the right one (or at least a slight modification of it). On the other end this definition have always seemed to be the most natural one, because historically category theory develop in the context of algebraic topology, so now I've a new question:

Does anyone know the logical process that took Mac Lane and Eilenberg to give their (classical) definition of natural transformation?
Here I'm interested in the topological/algebraic motivation that move those great mathematicians to such definition rather the other one.

REPLY [4 votes]: Following the previous indication of Professor Brown I want to add another possible way to see natural transformation which is a generalization of the previous definition.

Given categories $\mathcal C$ and $\mathcal D$ and two functors between them $\mathcal F,\mathcal G \colon \mathcal C \to \mathcal D$ then a natural transformation $\tau$ can be defined as a functor $\tau \colon \mathcal C \to (\mathcal F \downarrow \mathcal G)$ which arrow components are the diagonal functions, sending each arrow $f \in \mathcal C(c,c')$, with $c,c' \in \mathcal C$ to $(f,f) \in (\mathcal F \downarrow \mathcal G)(\tau(c),\tau(c'))$.

Edit: I think the definition of natural transformation proposed by professor Brown probably can be even a more natural than the one proposed in the question. 
I think that more details are worthed. 
The key ingredient for that definition is the concept of arrow category of a given category $\mathbf D$: such category have morphism of $\mathbf D$ as objects and commutative square as morphisms. 
This category come equipped with two functors $\mathbf {source}, \mathbf{target} \colon \text{Arr}(\mathbf D) \to \mathbf D$ such that for each object (i.e. a morphisms of $\mathbf D$) $f \colon d \to d'$ we have 
$$\mathbf{source}(f)=d$$
$$\mathbf{target}(f)=d'$$
while for each $f \in \mathbf D(x,x')$, $g \in \mathbf D(y,y')$ and a morphism $\alpha \in \text{Arr}(\mathbf D)(f,g)$ (i.e. a quadruple $\langle f,g, \alpha_0,\alpha_1\rangle$ where $\alpha_0 \in \mathbf D(x,y)$ and $\alpha_1 \in \mathbf D(x',y')$ such that $\alpha_1 \circ f = g \circ \alpha_0$) we have 
$$\mathbf{source}(\alpha)=\alpha_0$$
$$\mathbf{target}(\alpha)=\alpha_1$$
it's easy to prove that these data give two functors (which gives to $\text{Arr}(\mathbf D)$ the structure of a graph internal to $\mathbf{Cat}$).
Now let's take a look to this new definition of natural transformation:

A natural transformation $\tau$ between two functors $F,G \colon \mathbf C \to \mathbf D$ is a functor $\tau \colon \mathbf C \to \text{Arr}(\mathbf D)$ such that $\mathbf{source} \circ \tau = F$ and $\mathbf{target}\circ \tau = G$.

A functor of this kind associate to every object $c \in \mathbf C$ a morphism $\tau_c \colon F(c) \to G(c)$ in $\mathbf D$, while to every $f \in \mathbf C(c,c')$ it gives the commutative triangle expressing the equality 
$$\tau_{c'} \circ F(f)=\tau_{c'} \circ \mathbf {source}(\tau_f)=\mathbf {target}(\tau_f) \circ \tau_c = G(f) \circ \tau_c$$
certifying the naturality (in the ordinary sense) of the $\tau_c$.
This definition reminds the notion of homotopy between maps $f,g \colon X \to Y$ as map of kind $X \to Y^I$ (i.e. an homotopy as a (continuous) family of path of $Y$). 
That's not all, indeed we can reiterate the construction of the arrow category obtaining what I think is called a cubical set
$$\mathbf D \leftarrow \text{Arr}(\mathbf D) \leftarrow \text{Arr}^2(\mathbf D)\leftarrow \dots $$
where each arrow should be thought as the pair of functors $\mathbf{source}_{n+1},\mathbf{target}_{n+1} \colon \text{Arr}^{n+1}(\mathbf D) \to \text{Arr}^n (\mathbf D)$.
In this way we can associate to each category a cubical set. There's also a natural way to associate to every functor a (degree 0) mapping of cubical sets.
If we consider natural transformation as maps from a category to an arrow category then this correspondence associate to each natural transformation a degree 1 map between such cubical sets (by degree one I mean that the induced map send every object of $\text{Arr}^n(\mathbf C)$ in an object of $\text{Arr}^{n+1}(\mathbf D)$).
I've found really beautiful this construction because it shows an analogy between categories-functors-natural transformation and complexes-map of complexes-complexes homotopies.