I think what you have defined is just called the **category of endomorphisms** of $\mathcal{C}$. See for instance Marian Mrozek's 1992 [paper][1] *Normal functors and retractors in categories of endomorphisms* for some properties of this category.

The oldest mention that I have found in the literature so far is  

> **Almkvist, Gert**, *The Grothendieck ring of the category of endomorphisms*, J. of Algebra 28 (1974), 375–388.

Almkvist restricts attention to the case where $\mathcal{C}$ is the category of $A$-modules which admit a finite projective resolution. See also [this question][2] of mine about homotopy properties of iterating the endomorphism construction.


  [1]: http://www2.im.uj.edu.pl/actamath/PDF/29-181-198.pdf
  [2]: http://mathoverflow.net/questions/136195/how-does-it-end