Injective objects in a category

Question

Let $C$ be a category and $x\in$ Ob$(C)$. Assume that $I\in$ Ob$(C)$ is injective in $C$, and there are morphisms $f:x\rightarrow I,~ g:I\rightarrow x$ such that $gf =1_x$. Is it true that $x$ is injective as well? Is there any common conditions to force such a fact to be true? Thanks!

Answer 1

I think so. Take a monomorphism $h: A\hookrightarrow B$ and any morphism $k: A\rightarrow x$. Then $fk$ factors through $h$ and postcomposing with $g$ gives the desired factorization of $k$.