# Naturality (part 2)

February 8, 2012 Leave a comment

In a category with products, and objects , , and , the objects and are technically not the same object, but there’s an obvious isomorphism between them. But this is more than just your standard isomorphism. It’s not like we construct the isomorphism one way if we use a specific object , and a different way for another object . It doesn’t really depend on the objects , , or at all. It depends on the structure of the functor . How can we encode that?

We encode that by saying that the maps between objects and can be “puffed up” to maps between and , and to maps between and . Moreover, these puffed up maps are somehow not very different, in the sense that the way we “transform” from to is “the same” on the domain and codomain ends of the “puffed up” maps. Take some time to parse my attempt at writing a coherent sentence.

Take a look at the diagram again and see if you can understand what I was trying to get across. Here, is the functor and is the functor . Can you write down what should be? Does the diagram commute?