Are coslice categories of preadditive categories preadditive? I do not see any natural way how a coslice category of a preadditive category can be preadditive (other than in some degenerate cases). However, they are given as an example in Popescu's book "Abelian Categories with Applications to Rings and Modules" (1973), page 17:



*Let $\mathcal(C)$ be a preadditive category and let $X$ be an object of $\mathcal{C}$. We denote by $X/\mathcal{C}$ the category whose objects are couples $(f, Y)$, $Y\in\operatorname{Ob}{\mathcal{C}}$ and $f\in\operatorname{Hom}_{\mathcal{C}}(X, Y)$ and whose morphisms $g\colon (f, Y)\to(f', Y')$ are in fact morphisms $g\colon Y\to Y'$ such that $gf = f'$. $X/\mathcal{C}$ is a preadditive category. Analogously, we have the category $\mathcal{C}/X$.


Is this an error in the book, or am i missing something?

You are right and it is easy to come up with examples. In fact, every non-trivial example does it: Suppose that $C$ has a non-initial object $X$. Then there exists an object $Y$ with a non-zero morphism $f'\colon X\to Y$. Moreover, let $f\colon X\to Y$ be the zero morphism (i.e., the unit of the abelian group $\hom(X,Y)$). Then there is no $g\colon Y\to Y$ such that $gf=f'$ since the left-hand side is the zero-morphism and the right hand side isn't. But then $\hom((Y,f),(Y',f'))=\emptyset$ cannot be an abelian group.
Thus, the only case that works is if $X$ is initial, but then we have $X/C\cong C$.
