Name for a category Is there any name or notation for this category?
Let $U$ be a set. By "function" I will mean a function $U\rightarrow U$.


*

*objects are functions;

*morphisms from a function $A$ to a function $B$ are such functions $f$ that $y=A(x)\Rightarrow f(y)=B(f(x))$ for every $x,y\in U$.

 A: Let $\mathcal{U}$ be the category with a single object $U$ and having as morphisms all functions $U\to U$, and consider the monoid $(\mathbb{N},+)$ as a category in the standard way. Then the category you described can be viewed as the functor category $Funct(\mathbb{N},\mathcal{U})$. Any functor $\mathbb{N}\to\mathcal{U}$ is arises from and is determined by where it takes the arrow generating the monoid $\mathbb{N}$, so you can think of the objects of this category as functions $U\to U$. Any natural transformation between two such functors (i.e. functions $A$ and $B$) arises from and determines a function $f\colon U\to U$ satisfying the equation you asked for. 
You can get quite similar categories as functor categories of some kind or another. For example, if $\mathcal{C}$ is the category with two objects $A$ and $B$ and one non-trivial arrow $f\colon A\to B$, then $Funct(\mathcal{C},\mathrm{Set})$ is a category where the objects are essentially functions between sets and morphisms are pairs of functions making the obvious square commute.
