Is there a nontrivial abelian category $C$ such that both $C$ and $C^{op}$ satisfy AB3-AB5? ("nontrivial" means that there are nonzero objects in $C$)



Abelian category which is nice and its dual is also
The answer is no.


This is mentioned (with a hint for a proof) on page 129 of A. Grothendieck, *Sur quelques points d'algèbre homologique,* Tohoku Math. J. (2) 9, (1957), 119-221.


For a complete proof (albeit in somewhat different terminology) see Proposition III.1.10 in B. Mitchell, *Theory of categories,* Academic Press (1965).


