Natural transformation

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

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

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

1. 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.

1. 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.