A vertical transformation is an analogue of a natural transformation which goes between double functors of double categories, and whose components are vertical arrows and squares. There is a dual notion of horizontal transformation.
If and are strict double categories regarded as internal categories in and are double functors regarded as internal functors in , then a transformation between them is simply an internal natural transformation in . Whether this is a vertical or horizontal transformation depends on how we identify double categories with internal categories in (there being two ways).
More explicitly, a vertical transformation consists of
For every object , a vertical arrow
in , which are natural with respect to vertical composition of vertical arrows in .
For every horizontal arrow in , a square
in , which are natural with respect to vertical composition of squares in .
For each , if is its horizontal identity, then the square is equal to , the identity square on the arrow .
For and , the horizontal composite of and is equal to .
The notion of horizontal transformation is dual.
Another characterization of transformations between double categories comes from observing that the 1-category is cartesian closed, and so any two double categories have an exponential . The objects of are double functors, its vertical arrows are vertical transformations, and its horizontal arrows are horizontal transformations. Its squares are a sort of “square modification” relating a pair of vertical and a pair of horizontal transformations.
It is easy to modify the explicit definition to handle the cases when and are weak in one direction or the other, and/or when and are pseudo functors in one direction or the other, by composing with appropriate coherence constraints. In this way, we obtain many 2-categories of double categories.
It is also easy to define vertical transformations between double functors which are horizontally lax or colax, and dually. In fact, given double categories , lax functors and , and colax functors and , we can define a vertical transformation having the shape
despite the fact that the composites and do not exist as double functors of any sort. Such transformations are the squares of a large double category whose objects are double categories, whose horizontal arrows are lax functors, and whose vertical arrows are colax functors. This, in turn, is a special case of a construction which works for algebras over any 2-monad.
Finally, we can also define vertical transformations between functors of (horizontally) virtual double categories.