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**xianghu21** Let **Tors **be the category whose objects are torsion abelian groups; if $\displaystyle A$ and $\displaystyle B$ are torsion abelian groups, we define $\displaystyle Mor_{\text{Tors}}(A, B)$ to be the set of all (group) homomorphisms $\displaystyle \phi : A \rightarrow B$. Prove that direct products exist in **Tors**; that is, show that given any indexed family $\displaystyle \{A_i\}_{i \in I}$ where $\displaystyle A_i$ is a torsion abelian group, there exists a torsion abelian group which serves as a direct product for this family in **Tors**.