Find the categorical sequential limit of $... \to \mathbb Q \to \mathbb Q \to \mathbb Q$ in the category of abelian groups, where all arrows are multiplication by a natural number $n$, with $n$ fixed and greater than $1$.
It suffices to check - since all limits are unique up to unique isomorphism - that $\mathbb Q$ works. Relabel the sequence as $... \to \mathbb Q_3 \to \mathbb Q_2 \to \mathbb Q_1$ However, I am not exactly sure how I would do that. Given an abelian group $X$, we have to contruct a set of maps $\phi_i$ such that the diagram defining the limit commutes (I won't post the full description of the definition of a limit because it's in wikipedia). The definition of a limit seems pretty technical to me. Is there an intuitive way to find a limit of a categorical sequence like this?
I am fairly new to category theory so please excuse me for this basic question.
Edit: I get that $\mathbb Q$ works as one that commutes, but how would I show thame universal property? I realize that, if $(X, \psi_i)$ is another cone then $\psi_i$'s are comoletely determined by $\psi_1$.
If you label the $\Bbb Q$s as you have done, counting from the right, the map from the limit (which is $\Bbb Q$ as you aver) to the $k$-th $\Bbb Q$ can be chosen to be $x\mapsto n^{-k}x$.