# What is this name of this 2-category without very much structure?

I was wondering if there is a name for this 2-category which is like the 2-category of natural transformations, but does not actually require the 1-morphisms to be functors or the 2-morphisms to be natural transformations. That is

• the objects are categories
• the 1-morphisms between $\mathcal{C}$ and $\mathcal{D}$ are functions $Ob(\mathcal{C}) \to Ob(\mathcal{D})$
• the 2-morphisms between $F, G : \mathcal{C} \to \mathcal{D}$ are assignments $\eta_c$, for each object $c$ of $\mathcal{C}$, to a morphism $Fc \to Gc$

For my application I am typically considering the case where the objects are (category product) powers of a single category, rather than all categories, but I suspect it does not make much difference.

• Isn't this the category of natural transformation from $\mathcal{C}^d$ to $\mathcal{D}$ where $\mathcal{C}^d$ is the "discretization" of $\mathcal{C}$ (i.e. the category with the same objects but only the identity morphisms)? Feb 7 '14 at 18:37

This is not a 2-category: there is no way to compose a 2-morphism $\eta : F\to G : C\to D$ with a function $H:Ob(D)\to Ob(E)$.