I'm interested in a problem involving finite groups, where I've discovered that inverse systems and their limits are useful for describing certain subgroups. I'm especially interested in the inverse system of finite groups over a finite poset because its limit is also finite.
My question is as follows: Consider two finite posets, $I$ and $J$, and let $G_{(i,j)}$ be a finite group for every pair $(i,j)$ in $I\times J$. We have an inverse system denoted by $((G_{(i,j)})_{(i,j)\in I\times J}, (f_{(i_1,j_1)(i_2,j_2)})_{(i_1,j_1)\le (i_2,j_2)\in I\times J})$. I would like to know if the following formula holds true: $$\lim_{i\in I}\lim_{j\in J}G_{(i,j)}\cong\lim_{(i,j)\in I\times J}G_{(i,j)}\cong \lim_{j\in J}\lim_{i\in I}G_{(i,j)}$$
@Käsekuchen gave a wonderful comment on this question.
I'm not very familiar with category theory. I would greatly appreciate it if you could provide me with a reference for above questions.