The comma category of two functors and is like an arrow category of where all arrows have their source in the image of and their target in the image of (and the morphisms between arrows keep track of how these sources and targets are in these images). It is a kind of 2-pullback: a directed refinement of the homotopy pullback of two functors between groupoids.
We discuss three equivalent definitions of comma categories
In terms of the imagery of loop spaces objects, the comma category is the category of directed paths in which start in the image of and end in the image of .
If and are both the identity functor of a category , then is the category of arrows in .
If is the identity functor of and is the inclusion of an object , then is the slice category .
Likewise if is the identity and is the inclusion of , then is the coslice category .
The comma category comes with a canonical 2-cell in the square
which is universal in the 2-categoryCat; that is, it is an example of a 2-limit (in fact, it is a strict 2-limit). Squares with the same universal property in an arbitrary 2-category are called comma squares and their top left vertex is called a comma object.