For any two objects (categories) \(C\) and \(D\) in Cat, the collection of morphisms (functors) from \(C\) to \(D\) form a category called a "functor category," written \([C,D]\). This is an exponential object in \(\texttt{Cat}\), so it might also be written \(D^C\). The objects in a functor category are functors from \(C\) to \(D\). The arrows are natural transformations, which transform one functor into another.

Similar to how functors are like functions with an extra restriction that makes them useful, natural transformations are transformations that satisfy a "naturality condition," hence calling them "natural." Formally, for two functors \(F,G:C\to D\), a natural transformation \(\alpha:F\Rightarrow G\) is a family of morphisms in \(D\) transforming "outputs" of \(F\) into "outputs" of \(G\). Namely, \(\alpha_X:FX\to GX\). The naturality condition that natural transformations must satisfy is depicted below as a diagram that must commute, and also illuminates how the transformation works:

That is, \(Gf\circ\alpha_X=\alpha_Y\circ Ff\). Note that this entire diagram takes place in \(D\), and the only mention of \(C\) are \(X\) and \(Y\), which are objects in \(C\).

© 2024 Ryan Brewer.