TITLE: inverse limit isomorphisms
QUESTION [7 upvotes]: Let $(I,\leq )$, $(J,\leq )$ be partially ordered sets and $G_{ij}$ be a set of groups indexed by pairs $(i,j)\in I\times J$.The product of the partially ordered sets $(I,\leq )$, $(J,\leq )$ is $I\times J$ with the partial order $(i,j)\leq (i',j')$ iff $i\leq i'$ and $j\leq j'$. Suppose that for all $(i,j)\leq (i',j')$ we are given homomorphisms $f_{ij,i'j'}:G_{i'j'}\rightarrow G_{ij}$ satisfying the usual transitivity conditions. How to prove that there are canonical isomorphisms : $\varprojlim_{i}\varprojlim_{j}G_{ij}\sim \varprojlim_{(i,j)}G_{ij}\sim \varprojlim_{j}\varprojlim_{i}G_{ij}$?
Thank you.

REPLY [4 votes]: Instead of the category of groups you can work in any category. By the Yoneda Lemma it suffices to prove the statement in the category of sets. But there it is easily checked, since all three sets are the same subsets of $\prod_{i,j} G_{i,j}$. You have compatibility in $(i,j)$, iff you have compatibility in $i$ and compatibility in $j$: $\Rightarrow$ follows by considering identities in one entry, and $\Leftarrow$ follows by composition.