Why completeness alone cannot replace a solution set or the SAFT smallness hypotheses

Completeness supplies limits of small diagrams. It does not make the class of candidates in a comma category small, does not supply a jointly weakly initial set, and does not turn a proper collection of subobjects into a small diagram. Those are the roles of the solution-set condition in GAFT and the coseparating, well-powered, or explicit intersection-preservation data in SAFT.

The distinction disappears only under restrictive size hypotheses, and then only assuming Choice, which both of the following results carry as a hypothesis. Assuming Choice, a small complete category is forced toward preorder behaviour by Assuming Choice, every small complete category and every small cocomplete category is a preorder, and sufficiently large products or coproducts force the same conclusion by Assuming Choice, a small category with products or coproducts indexed by the cardinality of its morphism set is a preorder. Large-category claims here use the definable-class convention of Class-sized category theory in ZFC: definable-class schemas, small and locally small categories, and why $\mathbf{C}\mathbf{A}\mathbf{T}$ is not formed.