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The usual definition of hom-sets between ind-objects in a category $C$ is:

$$ \operatorname{ind-}C(F,G) := \lim{}_{a\in A} \operatorname{colim}_{b\in B} \operatorname{Hom}(Fa,Gb)\;. $$

where $F:A\to C$ and $G:B\to C$ are ind-objects, i.e. functors from filtered categories.

Now suppose that $A=B$.

Can we interchange the order of limit and colimit?

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    $\begingroup$ No. Just take any example of, say, a sequential limit and colimit not commuting, and then make those your Hom sets. E.g. Z/(p^k). Take colimit first then limit, you get zero. Take limit first then colimit and you get Q_p. $\endgroup$
    – Dylan Wilson
    Commented Dec 17, 2016 at 23:25
  • $\begingroup$ @DylanWilson: This is not a comment. $\endgroup$
    – HeinrichD
    Commented Dec 18, 2016 at 22:19
  • $\begingroup$ upload.wikimedia.org/wikipedia/en/b/b9/MagrittePipe.jpg $\endgroup$
    – Dylan Wilson
    Commented Dec 19, 2016 at 3:23

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