= Natural transformation
{title2=$\alpha:F\Rightarrow G$}
{wiki}
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> $A$, with
$$
G(f)\circ\alpha_A=\alpha_B\circ F(f)\qquad\text{for every }f:A\to B.
$$
This <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> $f:X\to Y$,
$$
\mathcal P(f)(\eta_X(x))=f(\{x\})=\{f(x)\}=\eta_Y(f(x)).
$$
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>.
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