Example of non-filtered colimit not commuting with a finite limit

Question

I know that the canonical morphism $\mathrm{Colim}_i\mathrm{Lim}_jD(i,j)\to\mathrm{Lim}_j\mathrm{Colim}_iD(i,j)$ for a diagram $D:\textbf{I}\times\textbf{J}\to \textbf{Set}$ is not in general an isomorphism, but it is in the case in which the categories $\textbf{I}$ and $\textbf{J}$ are respectively filtered and finite.

I'm able to think about counterexamples for $\textbf{I}$ filtered (and of course $\textbf{J}$ infinite), and I would like to find another counterexample in the case of $\textbf{J}$ finite (and $\textbf{I}$ non-filtered).

Thanks in advance for any suggestion!

Answer 1

You can take $I=J$ be discrete categories with just two objects each. Then nearly any diagram $I\times J \to \mathbf {Set}$ will produce a morphism that is not an isomorphism.