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

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