I have a certain construction which I believe have been studied before, I just don't know where to look
Let $C, D$ be categories and $U: D \to C$ a functor The category of $U$-extensions is the following: Objects: pairs $(F, \eta)$ where $F:C\to D$ is a functor and $\eta: id_{End(C)}\Rightarrow UF$ is a natural transformation
Morphisms $(F_1, \eta_1) \to (F_2, \eta_2)$ are natural transformations $\xi:F_1\Rightarrow F_2$ commuting with the eta's: $\xi \eta_1 = \eta_2$
What is the real name of this category, what has been investigated about it?