= Solution
A <direct system of abelian groups> over the directed poset $A$ consists of groups $G_a$ and maps $\phi_{ab}:G_a\to G_b$ for $a\leq b$, satisfying $\phi_{aa}=1$ and $\phi_{bc}\phi_{ab}=\phi_{ac}$. Its <direct limit> is the quotient of $\bigoplus_aG_a$ by the relations $g_a\sim\phi_{ab}(g_a)$.
For the displayed sequence, put $P_0=1$ and $P_n=a_0a_1\cdots a_{n-1}$. Map the copy of $\mathbb Z$ at stage $n$ to $\mathbb Q$ by
$$
m\longmapsto \frac{m}{P_n}.
$$
This is compatible with the next transition because $a_nm/P_{n+1}=m/P_n$. The universal property of the <direct limit> therefore identifies it with
$$
\boxed{\bigcup_{n\geq0}\frac1{P_n}\mathbb Z.}
$$
In reduced form, these are exactly the rationals whose denominator divides one of the finite products $P_n$. This is the <sequential direct limit of multiplication maps on the integers>.
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