Natural transformation ($\alpha :F\Rightarrow G$)

A natural transformation between functors $F,G:\mathcal{C}\to \mathcal{D}$ assigns a categorical morphism ${\alpha }_A:F(A)\to G(A)$ to every object of a category $A$, withThis naturality condition says that transforming the input before transporting it gives the same result as transporting it first and transforming the output. It constrains the whole family of arrows, so unrelated choices of a map for each object generally fail it.

For a concrete example, compare the functor that leaves every set and function unchanged with the power set functor using images of subsets. The maps ${\eta }_X:X\to \mathcal{P}(X)$ given by $x\mapsto \{x\}$ form a natural transformation: for any function $f:X\to Y$,The right-hand panel of the category illustration checks the square at $x=3$ with $f(x)=x+1$: both routes end at $\{4\}$. The statement works for every function, rather than depending on a preferred labeling of a particular set.

For $F,G:{\mathcal{C}}^{\mathrm{op}}\times \mathcal{C}\to \mathcal{D}$, a dinatural transformation has components ${\alpha }_C:F(C,C)\to G(C,C)$ satisfying, for $f:C\to D$,Both composites have domain $F(D,C)$ and codomain $G(C,D)$. It is the compatibility used for categorical ends and categorical coends, rather than ordinary naturality on just one variance.

A natural transformation all of whose components are isomorphisms. Their inverses form the inverse natural transformation.

A natural bijection between set-valued functors is a natural transformation whose component functions are bijections. Their inverse functions are automatically natural, so it is a natural isomorphism.

For a natural transformation $\theta :F\Rightarrow G$, naturality is the equation $G(f){\theta }_A={\theta }_{B}F(f)$ for each morphism $f:A\to B$. Thus transforming and then applying a functor gives the same result as applying that functor and then transforming. It makes the component maps a coherent transformation of functors.