Initial object criterion for a covariant local topos ($[\mathcal{C},\mathbf{Set}]\text{ local}⟺\mathrm{Kar}(\mathcal{C})\text{ has an initial object}$)

The constant singleton functor must be a retract in a category of a covariant representable functor. Necessity follows by applying a colimit-preserving global-sections functor to the canonical coproduct of representables onto the terminal functor. Conversely its Hom functor is a retract of evaluation, preserves colimits and has a right adjoint. Such a retract is represented by an initial object in the idempotent completion. For an idempotent-complete category an actual initial object suffices; without this hypothesis an absorbing-zero monoid gives a counterexample to the unqualified claim.