Let $\mathcal C$ be a category, and consider a new category $\mathcal C'$ with


$Obj(\mathcal C') := \{$pairs $(X \in Obj(\mathcal C), T \in End\_{\mathcal C}(X)) \}$


$Hom\_{\mathcal C'}((X,T\_X),(Y,T\_Y)) := \{R \in Hom\_{\mathcal C}(X,Y) : T\_Y \circ R = R\circ T\_X \}$



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> Does this $\mathcal C'$ have a name, or any non-formally-obvious properties? 
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I suppose that the more natural construction would let $T$ be an arbitrary morphism instead of an endomorphism. In my context $\mathcal C$ is additive and I require also that $T^2=0$.



Slice-category-like terminology question
I think what you have defined is just called the **category of endomorphisms** in $\mathcal{C}$. See for instance Marian Mrozek's 1992 [paper](http://www2.im.uj.edu.pl/actamath/PDF/29-181-198.pdf) *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 



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> **Almkvist, Gert**, *The Grothendieck ring of the category of endomorphisms*, J. of Algebra 28 (1974), 375–388.
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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](https://mathoverflow.net/questions/136195/how-does-it-end) of mine about homotopy properties of iterating the endomorphism construction.


