Category of ordinals in reverse order

Regard the proper class of all ordinals as a poset category with $\alpha \to \beta$ when $\alpha \ge \beta$ in the usual order. It is locally small and complete: the product of any set-indexed family is its ordinary supremum, and equalizers are automatic in a poset. It has no initial object because there is no largest ordinal. Consequently every representable functor from it to sets preserves all small limits but has no left adjoint, since a left adjoint would have to send the empty set to an initial object.