Are monomorphisms and epimorphisms closed under addition and scalar factor?

Question

In a preadditive category, i.e. each hom-set has an abelian group structure, is it true that if $f,g:X\to Y$ are monomorphisms/epimorphisms, then so are $-f$, $-g$, and $f\pm g$? In particular, if $C$ is an abelian category, then $-id_X$ is an isomorphism?

Answer 1

In every preadditive category, $\newcommand\id{\operatorname{id}}-\id_X:X\to X$ is an isomorphism for every object $X$ because $(-\id_X)\circ(-\id_X)=\id_X$. This follows from $-\id_X+\id_X=0_X$ by composing with $-\id_X$ which gives $(-\id_X)\circ(-\id_X)+(-\id_X)=0_X$, hence \begin{align} \id_X &=0_X+\id_X\\ &=(-\id_X)\circ(-\id_X)+(-\id_X)+\id_X\\ &=(-\id_X)\circ(-\id_X)+0_X\\ &=(-\id_X)\circ(-\id_X) \end{align} Let $f:X\to Y$ be a morphism. Since $$(-f)\circ(-\id_X)=f=(-\id_Y)\circ(-f)$$ $-f:X\to Y$ is monic/epic/iso if and only if $f$ is monic/epic/iso.

On the other hand, as pointed out in comments, $f\pm g$ can fails to be monic/epic/iso even when both $f$ and $g$ are monic/epic/iso.