Definition and notation
Let be a category. A square of morphisms of consists of objects of and morphisms , , , and . This is often pictured as a square
The square is commutative if .
The class of commutative squares in is written .
This class has partial compositions and which are vertical and horizontal:
thus forming a (strict) double category, also written . It contains the vertical category and the horizontal category .
One can also form multiple compositions of arrays , , of commutative squares provided that in the obvious sense adjacent squares are composible. One checks by induction that:
any composition of commutative squares is commutative.
Let denote the walking arrow: the category with two objects and one arrow . This has the structure of cocategory. Then the class of commutative squares in can also be described as .
If is a category, then can be regarded as the class of natural transformations of functors . Then the category structure induces a category structure on giving the functor category : the category of functors and natural transformations. (This account is due to Charles Ehresmann.)
One deduces that if also is a category then there is a natural bijection
which thus states that the category of (small if you like!) categories is cartesian closed.
The commutative squares serve as the morphisms in the arrow category of , which is the functor category .