Sequentially small object ($\mathrm{colim}\mathrm{Hom}(A,Z_k)\cong \mathrm{Hom}(A,\mathrm{colim}{Z}_k)$)

Mapping out of the object preserves sequential colimits. A map factors through a finite stage, and two finite-stage maps agreeing in the colimit agree at a later stage. Relative sequential smallness restricts the diagrams to a stated class of morphisms.