Natural Transformation

In Category Theory, a natural transformation is an arrow between pairs of functors, making it a 2-morphisms in the category Cat. Given$𝓒,𝓓:\mathbf{Cat}$; $F,G:𝓒\to 𝓓$ functors, A natural transformation $\alpha :F\Rightarrow G$ is a family of functors $\alpha :{\prod }_{X:\mathcal{C}}FX\to GX$ such that, for all pairs $a,b\in 𝓒$, and arrows $f:a\to B$ the following commutes:

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
FA \arrow[d, "Ff"] \arrow[r, "\alpha_A"] & GA \arrow[d, "Gf"] \\
FB \arrow[r, "\alpha_B"] & GB
\end{tikzcd}
\end{document}

All types in languages like haskell are natural transformations.

Duplicate Content 1

Let $\mathcal{C},\mathcal{D}$ be categories. Let $F,G:\mathcal{C}\Rightarrow \mathcal{D}$ be functors. A natural transformation $\mathcal{N}:F\Rrightarrow G$ is defined as a family of morphisms ${\alpha }_x:Fx\to Gx$ for every object $x\in \mathcal{C}$ such that the following ‘naturality square’ commutes:

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
F x \arrow[d, "\alpha_x"'] \arrow[r, "Ff"] & F y \arrow[d, "\alpha_y"] \\
G x \arrow[r, "Gf"'] & G y
\end{tikzcd}
\end{document}