Generalizing a statement about direct limits in the category of $A$-modules to other categories

Question

The original problem is from Atiyah and Macdonald's Introduction to Commutative Algebra, Ex 2.15:

Suppose $(M_\alpha,\mu_{\beta\gamma})$ is a direct system of modules over a unital commutative ring $A$, $M=\varinjlim M_\alpha$, and $\mu_\alpha\colon M_\alpha\to M$ are natural maps. Given that $\mu_\alpha(x_\alpha)=0$, then there exists $\beta\ge\alpha$ such that $\mu_{\alpha\beta}(x_\alpha)=0$.

Also see: Does the following property of the direct limit of a direct system follow from the axioms for a direct limit?

Since I don't know category theory, maybe this one is a baby question. I wonder whether the statement could be generalized, for example, in abelian categories instead of the category of modules? I did try only to make use of universal mapping property to characterize the direct limit, but failed. The question should be a restricted version of usergen's question.

Any idea?