Natural transformation

Let $\mathcal C,\mathcal D$ be categories , and let $F,G:\mathcal C\to\mathcal D$ be functors .

Definition

A natural transformation $\eta:F\Rightarrow G$ assigns to every object $X\in\mathcal C$ a morphism

in $\mathcal D$, such that for every morphism $f:X\to Y$ in $\mathcal C$, the naturality condition

holds (composition in $\mathcal D$; see composition ).

Equivalently, for each $f:X\to Y$ the square commutes:

The components $\eta_X$ are called the components of $\eta$.

Examples

1. Singleton map $X\to\mathcal P(X)$ (Set).
   Let $\mathbf{Set}$ be the category of sets .
   Define the covariant “power set” functor $P:\mathbf{Set}\to\mathbf{Set}$ by $P(X)=\mathcal P(X)$ and, for a function $f:X\to Y$, let $P(f):\mathcal P(X)\to\mathcal P(Y)$ be the image map $S\mapsto f(S)$.
   Then the family $\eta_X:X\to\mathcal P(X)$, $x\mapsto\{x\}$, defines a natural transformation

   $$\eta:\mathrm{Id}_{\mathbf{Set}}\Rightarrow P,$$

   since $P(f)(\{x\})=\{f(x)\}$.

2. Postcomposition induces a natural transformation on representables.
   In any $\mathcal C$, fix a morphism $h:A\to B$. Consider the contravariant hom-functors $\mathrm{Hom}_{\mathcal C}(-,A)$ and $\mathrm{Hom}_{\mathcal C}(-,B)$ (see contravariant functor ).
   Define, for each $X$,

   $$\eta_X:\mathrm{Hom}_{\mathcal C}(X,A)\to \mathrm{Hom}_{\mathcal C}(X,B),\quad f\mapsto h\circ f.$$

   Naturality follows from associativity of composition.

3. The evaluation map into the double dual (Vect$_k$).
   In the category of $k$-vector spaces, there is a natural transformation

   $$\mathrm{ev}:\mathrm{Id}\Rightarrow (-)^{\ast\ast}$$

   whose component at $V$ is the canonical linear map $V\to V^{\ast\ast}$, $v\mapsto(\varphi\mapsto\varphi(v))$.
   For finite-dimensional $V$ this component is an isomorphism, so $\mathrm{ev}$ becomes a natural isomorphism on the full subcategory of finite-dimensional spaces.