I am trying to prove the following fact that given $I$ filtered index, $J$ finite index and diagram $F:I\times J \rightarrow \it{Sets}$, $colim_{i\in I} lim_{j\in J}\;F_{ij}=lim_{j\in J}colim_{i\in I}\; F_{ij}$; provided all the corresponding limits and co-limits exist.
Now, I have considered $F_{j}= colim_{i\in I}\;F_{ij} $ for fixed $j\in J$ and $F_{i}= lim_{j\in J} \; F_{ij}$ for fixed $i\in I$. And I'm trying to find maps $F_{i}\rightarrow lim_{j\in J}\;F_{j}$. However I'm completely clueless about how to use the finite index condition.
I can't seem to find a proof of this anywhere and am stuck for some time. So, I would like to see a detailed proof if possible. Thanks in advance!
You don't need anything about finiteness or filteredness at this stage. You just compose the projection $F_i\to F_{ij}$ with the injection $F_{ij}\to F_j$ and check that this passes to the limit over $j$.