Given two functors \(F:C\to D\) and \(G:C\to D\), a "natural isomorphism" between them \(\alpha: F\cong G\) is an isomorphism in the functor category \([C,D]\). Then \(F\) and \(G\) are called "naturally isomorphic."
In other words, there are two natural transformations \(\alpha_\to: F\to G\) and \(\alpha_\leftarrow:G\to F\) such that \(\alpha_\to\circ\alpha_\leftarrow=id_G\) and \(\alpha_\leftarrow\circ\alpha_\to=id_F\). That is, the natural transformation (either of them, doesn't matter which) is reversible. Note that \(\circ\) here is referring to the typical (vertical) composition of a functor category, not horizontal composition.
It's worth considering what this looks like when we write the natural transformations in their "pointwise" form, as \(\alpha_{\to,X}:FX\to GX\) and \(\alpha_{\leftarrow,X}:GX\to FX\). The equations become \(\alpha_{\to,X}\circ\alpha_{\leftarrow,X}=id_G X\) and \(\alpha_{\leftarrow,X}\circ\alpha_{\to,X}=id_F X\). Naturally, natural isomorphisms have a pointwise notation as well: \(\alpha_X:FX\cong GX\). In all of these cases, the index \(X\) refers to objects in the category \(D\), the codomain of the functors.
© 2024 Ryan Brewer.