So we know that — the collection of all categories — forms a 2-category with functors as 1-morphisms and with natural transformations as 2-morphisms. It shouldn’t surprise us, then, that the collection of all categories enriched over a monoidal category also comprises a 2-category.
We need two concepts to make this go through: an “enriched” notion of a functor, and an “enriched” notion of a natural transformation. As we might expect, both of them will just be written out like the familiar notions of functor and natural transformation, but substituting the new monoidal structure for the cartesian monoidal structure of .
First of all, a functor from category to is a function , along with functions for each hom-set. So to define a -functor we keep the function on objects and replace the functions between hom-sets with morphisms between hom-objects. Of course, these must preserve compositions and identities, as encoded in the following diagrams:
which by now should look very familiar.
A natural transformation between two functors from to picks out a morphism in for each object in , subject to a “naturality” condition. To find an analogue of picking out a morphism from a hom-set we use the same trick we did for the identity: we pick a morphism from to a hom-object. That is, a -natural transformation consists of an -indexed family of arrows , which make the following diagram commute:
You should try to write this diagram out in the case of to verify that it becomes the familiar naturality square in that context.
Now the exact same constructions we used to compose natural transformations “vertically” and “horizontally” apply to -natural transformations, and the same arguments we used in the case of apply to give a 2-category of categories, functors, and natural transformations, all enriched over the monoidal category .