# Definition:Natural Transformation

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## Definition

Let $\mathbf C$ and $\mathbf D$ be categories.

### Covariant Functors

Let $F, G : \mathbf C \to \mathbf D$ be covariant functors.

A natural transformation $\eta$ from $F$ to $G$ is a mapping on $\mathbf C$ such that:

$(1): \quad$ For all $x \in \mathbf C$, $\eta_x$ is a morphism from $\map F x$ to $\map G x$.
$(2): \quad$ For all $x, y \in C$ and morphism $f: x \to y$, the following diagram commutes:
$\xymatrix{ \map F x \ar[d]^{\eta_x} \ar[r]^{\map F f} & \map F y \ar[d]^{\eta_y} \\ \map G x \ar[r]^{\map G f} & \map G y}$

### Contravariant Functors

Let $F, G: \mathbf C \to \mathbf D$ be contravariant functors.

A natural transformation $\eta$ from $F$ to $G$ is a mapping on $\mathbf C$ such that:

$(1): \quad$ For all $x \in \mathbf C$, $\eta_x$ is a morphism from $\map F x$ to $\map G x$.
$(2): \quad$ For all $x, y \in C$ and morphism $f: x \to y$, the following diagram commutes:
$\xymatrix{ \map F x \ar[d]^{\eta_x} & \map F y \ar[d]^{\eta_y} \ar[l]^{\map F f} \\ \map G x & \map G y \ar[l]^{\map G f} }$

## Historical Note

The concept of a natural transformation was introduced by Samuel Eilenberg and Saunders Mac Lane.

It came about during their investigations into vector spaces and their duals.