Consider the category $\mathsf{FinAb}$ of finite abelian groups. The structure theorem tells us that we can write down a skeleton for this category (a set of representatives for the isomorphism classes), which consists of the groups $\mathbb{Z}/n_1 \oplus \cdots \oplus \mathbb{Z}/n_s$ with positive integers satisfying $n_1 | \cdots | n_s$. But this does not tell us anything about the morphisms between finite abelian groups.
Question. Can we improve the structure theorem in such a way that we find an explicit and easy to describe (this also includes the morphisms), countable category $\mathcal{C}$ together with an equivalence of categories $\mathcal{C} \cong \mathsf{FinAb}$?
Of course, "the full subcategory consisting of all $\mathbb{Z}/n_1 \oplus \cdots \oplus \mathbb{Z}/n_s$" is no answer.
Variants:
a) There is an equivalence of categories $\mathsf{FinAb} \cong \mathsf{FinAb}^{op}$, given by $\hom(-,\mathbb{Q}/\mathbb{Z})$. We could require that $\mathcal{C}$ also comes equipped with an explicit anti-equivalence and that $\mathcal{C} \cong \mathsf{FinAb}$ is compatible.
b) Actually $\mathsf{FinAb}$ is a symmetric monoidal (abelian) category with the usual tensor product $\otimes_{\mathbb{Z}}$ (without unit). It would be great if we can set up an equivalence of symmetric monoidal categories $\mathcal{C} \cong \mathsf{FinAb}$; in particular $\mathcal{C}$ should be symmetric monoidal.
c) Probably everything may be reduced to $\mathsf{FinAb}_p$, the category of finite abelian $p$-groups, where $p$ is a prime number.
d) I am also happy with the category of finitely generated abelian groups. This is the initial finitely cocomplete symmetric monoidal abelian category.