Is $\text{Hom}(A,X)+\text{Hom}(A,Y)\to\text{Hom}(A,X+Y)$ always surjective?

Question

Let $\mathsf C$ be the category of abelian groups. $A$, $Z$ two objects of $\mathsf C$, $X$, $Y$ two subgroups of $Z$. Then my question is:

For any homomorphism $f:A\to X+Y$, can we always find two homomorphisms $g:A\to X$ and $h:A\to Y$ such that $f=g+h$?

And What if we replace $\mathsf C$ by an arbitrary abelian category, and subgroups by subobjects?

Any idea will be appreciated.

Answer 1

Take $Z=\mathbb{Q}$, $X=\mathbb{Z}[\frac{1}{2}]$, $Y=\mathbb{Z}[\frac{1}{3}]$ and $A=X+Y=\mathbb{Z}[\frac{1}{6}]$.

Then there are no non-zero maps $A\to X$ or $A\to Y$, since $X$ and $Y$ contain no elements divisible by arbitrarily large powers of $6$. So the identity map $A\to X+Y$ is not the sum of maps from $A$ to $X$ and to $Y$.