Let $A$ and $B$ be finite abelian groups. A common multiple of $A$ and $B$ is a triple $(C,c_A,c_B)$ where $C$ is a finite abelian group and $c_A:\ A\ \hookrightarrow\ C$ and $c_B:\ B\ \hookrightarrow\ C$ are embeddings. My question is:
Does there exist a least common multiple of $A$ and $B$?
Here I mean least in the sense that if $(T,t_A,t_B)$ is another common multiple of $A$ and $B$, then there exists a group homomorphism $f:\ C\ \rightarrow\ T$ such that $t_A=f\circ c_A$ and $t_B=f\circ c_B$.
EDIT: I tried a few things. The most promising seems writing $A\cong\bigoplus(\Bbb{Z}/q\Bbb{Z})^{a_q}$ where $q$ ranges over all prime powers, and the same for $B$. Then I thought maybe $C=\bigoplus_q(\Bbb{Z}/q\Bbb{Z})^{c_q}$ might work, where $$c_{p^k}:=\max\{\sum_{i\geq k}a_{p^i},\sum_{i\geq k}b_{p^i}\}-\sum_{i>k}c_{p^i}.$$ But I am unable to prove it. Note that these definitions make sense because $a_{p^i}=b_{p^i}=0$ for $i$ sufficiently large because $A$ and $B$ are finite.
EDIT: As the comments made clear to me, it turns out that I mean least in a different sense. What I want is that if $(T,t_A,t_B)$ is another common multiple of $A$ and $B$, then there exists an embedding $C\ \hookrightarrow\ T$.
Write $A\cong \bigoplus_{j\in \mathbb{N}} (\Bbb{Z}/a_j\Bbb{Z})$ and $B\cong \bigoplus_{j\in \mathbb{N}} (\Bbb{Z}/b_j\Bbb{Z})$ where $a_{j+1} \mid a_{j}$ and $b_{j+1}\mid b_{j}$.
This is basically the structure theorem for finite abelian groups, just indexed the other way around, of course almost all $a_j,b_j$ will be $1$.
Then consider $\bigoplus_{j\in \mathbb{N}}(\Bbb{Z}/ \operatorname{lcm} (a_j,b_j)\Bbb{Z})$.