Implication Details
Claim: If a category has cartesian filtered colimits and has coproducts and is distributive, then it is infinitary distributive.
Proof: Each functor preserves finite coproducts and filtered colimits, hence all coproducts.
Claim: If a category has cartesian filtered colimits and has coproducts and is distributive, then it is infinitary distributive.
Proof: Each functor preserves finite coproducts and filtered colimits, hence all coproducts.