2 added 640 characters in body

This "geometric" definition is well-known to category-theorists. See for example this youtube video by the Catsters, which introduces natural transformations. It should be also well-known to algebraic topologists working with model categories. But I have to admit that there are few introductions to category theory which emphasize this definition of a natural transformation.

Remark that this fits into a more general framework: For every category $C$, there is an isomorphism $[I,C] \cong Arr(C)$, where $Arr(C)$ is the arrow category of $C$. In particular, $Arr([C,D]) \cong [I,[C,D]] \cong [C \times I,D]$.

On the other hand, the usual definition is more easy to work with. For example how do you define the composition of two natural transformations, say given by $\alpha : C \times 2 \to D, \beta : C \times 2 \to D$ with $\alpha(-,1) = \beta(-,0)$? Of course you can just write it down explicitly, but then you end up working with the usual definition. But instead, you could also use that $\alpha,\beta$ correspond to a functor on the amalgam $(C \times 2) \cup_C (C \times 2)$ of the inclusions $(-,1)$ and $(-,0)$, and compose with the natural functor $C \times 2 \to (C \times 2) \cup_C (C \times 2)$ which "leaves out the middle point".

1

This "geometric" definition is well-known to category-theorists. See for example this youtube video by the Catsters, which introduces natural transformations. It should be also well-known to algebraic topologists working with model categories. But I have to admit that there are few introductions to category theory which emphasize this definition of a natural transformation.

Remark that this fits into a more general framework: For every category $C$, there is an isomorphism $[I,C] \cong Arr(C)$, where $Arr(C)$ is the arrow category of $C$. In particular, $Arr([C,D]) \cong [I,[C,D]] \cong [C \times I,D]$.