nLab functor 2-category

Idea

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 $\mathcal{C}$ and $\mathcal{D}$ two 2-categories, their functor 2-category is the 2-category whose

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

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.

Related concepts

• functor category

• (infinity,1)-category of (infinity,1)-functors