This is a homework exercise, so please don't post full solutions to the question below.
Grothendieck (I believe) introduced several axioms an abelian category A voluntarily could satisfy. In particular, we have the two following:
AB5 A is cocomplete and filtered colimits of exact sequences are exact.
2 AB5* A is complete and filtered inverse limits of exact sequences are exact.
Then my problem is the following:
A.4.7 (Weibel) Show that if $A\neq 0$, then A cannot satisfy both axiom AB5 and AB5*. Hint: consider $\oplus A_i \to \prod A_i$.
I have stared at this problem for a long time but no ideas have (nor much intuition for filtered limits) been born. If anyone has any helpful explanations or wonderful hints, I'd be really grateful. Thanks.