# Category of all functors into a given category

For categories $$\mathbb{A}$$ and $$\mathbb{B}$$ I know that the functor category $$\mathbb{B}^{\mathbb{A}}$$ is a good way to model the functors from $$\mathbb{A}$$ to $$\mathbb{B}$$ (because the objects of $$\mathbb{B}^{\mathbb{A}}$$ correspond to the functors from $$\mathbb{A}$$ to $$\mathbb{B}$$). My question is, what if I want to model all functors into the category $$\mathbb{B}$$ (functors with any source category) ? In other words, is there a nice category which has the collection of all functors into $$\mathbb{B}$$ as its collection of objects ?

The slice category $$\mathbf{CAT}/\mathbb B$$ has as its objects pairs $$(\mathbb A, A \colon \mathbb A \to \mathbb B)$$ and as morphisms functors $$F \colon \mathbb A \to \mathbb A'$$ making the triangle commute (i.e. $$F \circ A = A'$$). One can also consider triangles that commute only up to natural isomorphism, which may be more appropriate in this case, since $$\mathbf{CAT}$$ is actually a 2-category. This is given by the (underlying 1-category) of the slice 2-category.