Natural Transformations

Let $C$ and $D$ be categories, and let

F, G : C \to D

be functors.

A natural transformation $η : F \Rightarrow G$ assigns to each object $X \in C$ a morphism in $D$ ,

η_{X} : F (X) \to G (X),

called the component at $X$ .

These components satisfy the naturality condition: for every morphism $f : X \to Y$ in $C$ , the following square commutes:

\begin{matrix} F (X) & \overset{F (f)}{\to} & F (Y) \\ η_{X} ↓ & ↓ η_{Y} \\ G (X) & \overset{G (f)}{\to} & G (Y) \end{matrix}

Equivalently,

G (f) \circ η_{X} = η_{Y} \circ F (f) .

Consequence: the 2-category Cat

With natural transformations included, categories, functors, and natural transformations do not merely form a category—they form a 2-category:

0-cells: categories
1-cells: functors
2-cells: natural transformations

This expresses the idea that one can compare not only objects and morphisms, but also functors between them.

Example

Let Man be the category whose objects are smooth manifolds and whose morphisms are smooth maps. Define two functors $F, G : Man \to Man$ as follows:

$F (M) = T M$ , the tangent bundle of $M$ .
For a smooth map $f : M \to N$ , define
$F (f) = T f : T M \to T N,$
the differential (pushforward) of $f$ .
$G (M) = M \times M$ .
For a smooth map $f : M \to N$ , define
$G (f) = f \times f : M \times M \to N \times N .$

There is a natural transformation $η : F \Rightarrow G$ , where for each manifold $M$ ,

η_{M} : T M \to M \times M,

is the map

η_{M} (v_{x}) = (x, x) .

This simply sends each tangent vector $v_{x} \in T_{x} M$ to the pair of its base point with itself.

Naturality check.
Given any smooth map $f : M \to N$ , the naturality square

\begin{matrix} T M & \overset{T f}{\to} & T N \\ η_{M} ↓ & ↓ η_{N} \\ M \times M & \overset{f \times f}{\to} & N \times N \end{matrix}

commutes, since for any $v_{x} \in T_{x} M$ ,

(f \times f) (η_{M} (v_{x})) = (f (x), f (x)) = η_{N} (T f (v_{x})) .

Thus,

(f \times f) \circ η_{M} = η_{N} \circ T f,

so $η$ is indeed a natural transformation between the tangent-bundle functor and the diagonal-pair functor.