Dear All,

I hope all is well.

If I may, morphisms of different categories preserve the structural essence(s) characteristic of the corresponding categories. All these structure-preserving morphisms of different categories can be represented as natural transformations and geometrically objectified as commutative squares. Do you see any value in studying categories with natural transformations / commutative squares as objects?

Unless I am mistaken, the categories of natural transformations that have been studied are categories with natural transformations as morphisms (hence the two compositions: horizontal and vertical, along with their Godement compatibility condition, among others). In thinking of natural transformation / commutative square as a generalization of one too many particular structure-respecting maps of various categories (in the category of categories), what added insight(s), often associated with generalization, can we expect to gain into: 'don't tear' (objectified as commutative square)?

I'd be truly grateful to you for any pointers / suggestions you may have.

Thanking you,
Yours respectfully,
posina