Another useful example of a category is a comma category. The term comes from the original notation which has since fallen out of favor because, as Saunders MacLane put it, “the comma is already overworked”.
We start with three categories , , and , and two functors and . The objects of the comma category are triples where is an object of , is an object of , and is an arrow in . The morphisms are pairs — with an arrow in and an arrow in — making the following square commute:
So what? Well, let’s try picking to be the functor sending the single object of to the object . Then let be the identity functor on . Now an object of is an arrow , where can be any other object in . A morphism is then a triangle:
Work out for yourself the category .
Here’s another example: the category . Verify that this is exactly the arrow category .
And another: check that given objects and in , the category is the discrete category (set) .