functor 2-category



The analog of functor category as categories are generalized to (strict or weak) 2-categories. There are various versions of this depending on how strict the functors and the transformations between them are. In general, for 𝒞\mathcal{C} and 𝒟\mathcal{D} two 2-categories, their functor 2-category is the 2-category whose

  1. objects are strict, pseudo, lax, or colax 2-functors from 𝒞\mathcal{C} to 𝒟\mathcal{D},

  2. 1-morphisms are strict, pseudo, lax, or colax natural transformations of 2-functors;

  3. 2-morphisms are modifications between these.

Such functor 2-categories are the hom-objects in various versions of the 3-category 2Cat.