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 $T^2=0$.