In exercise 5.34(iv) of Homological Algebra book by Rotman one is asked to prove that direct limits and inverse limits do not necessarily commute. I have two questions :
1.) Is it true that $\bigoplus \prod \mathbb{Z} \ncong \prod \bigoplus \mathbb{Z}$ (where the sum and product are over a countable index) ? How does one prove it ?
and
2.) How to find an example of a system $\{X_{i\alpha}\}$ such that $$\lim\limits_{{\rightarrow}_{i}} \lim\limits_{{\leftarrow}_{\alpha}} X_{i \alpha} \cong \lim\limits_{{\leftarrow}_{\alpha}} \lim\limits_{{\rightarrow}_{i}} X_{i \alpha}$$ but no isomorphism exists which would make the diagram
commute ?
I believe by saying that direct limits and inverse limits do not commute, Rotman means the requirement that no isomorphism making the above diagram commute exists. Am I understanding it correctly ? Thanks.