Slice-category-like terminology question Let $\mathcal C$ be a category, and consider a new category $\mathcal C'$ with
$Obj(\mathcal C') := \{$pairs $(X \in Obj(\mathcal C), T \in End_{\mathcal C}(X)) \}$
$Hom_{\mathcal C'}((X,T_X),(Y,T_Y)) := \{R \in Hom_{\mathcal C}(X,Y) : T_Y \circ R = R\circ T_X \}$

Does this $\mathcal C'$ have a name, or any non-formally-obvious properties? 

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 $T^2=0$.
 A: $\mathcal{C}'$ is isomorphic to a functor category, namely the category of functors $\mathbb{B} \mathbb{N} \to \mathcal{C}$, where $\mathbb{B} \mathbb{N}$ is the category freely generated by an endomorphism, i.e. the category with one object $*$ and $\mathrm{Hom}(*, *) = \mathbb{N}$.
If $\mathcal{C}$ is an $\mathbf{Ab}$-category and you also require $T^2 = 0$, then $\mathcal{C}'$ is isomorphic to the category of additive functors $\mathbb{B} A \to \mathcal{C}$, where $\mathbb{B} A$ is the $\mathbf{Ab}$-category with just one object $*$ and $\mathrm{Hom}(*, *) = A$, where $A$ is the ring $\mathbb{Z} [T] / (T^2)$. As always, $\mathcal{C}'$ is an $\mathbf{Ab}$-category.
A: I think what you have defined is just called the category of endomorphisms in $\mathcal{C}$. See for instance Marian Mrozek's 1992 paper 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 of mine about homotopy properties of iterating the endomorphism construction.
A: I would call it the category of representations of the loop quiver in $C$.
