Completing Categorical Algebras

Let Σ be a ranked set. A categorical Σ-algebra, cΣa for short, is a small category C equipped with a functor σC: C n →C, for each σ ∈ Σn, n ≥ 0. A continuous categorical Σ-algebra is a cΣa which has an initial object and all colimits of ω-chains, i.e., functors ℕ≥C; each functor σc preserves colimits of ω-chains. (ℕ is the linearly ordered set of the nonnegative integers considered as a category as usual.)