On definability of completely decomposable torsion-free Abelian groups by certain groups of homomorphisms

Let $C$ be an Abelian group. An Abelian group $A$ in some class $X$ of Abelian groups is said to be $_CH$-definable in the class $X$ if for any group $B\in X$ the isomorphism $\mathrm{Hom}(C,A)\cong\mathrm{Hom}(C,B)$ implies that $A\cong B$. If every group in $X$ is $_CH$-definable in $X$, then the class $X$ is called a $_CH$-class. In this paper we study conditions that make a class of completely decomposable torsion-free Abelian groups a $_CH$-class, where $C$ is a vector group.