A note on Noetherian (∞, ∞)-categories

The purpose of this note is to resolve a conjecture in arXiv:2307.00442(4), regarding the initial algebra for the enrichment endofunctor (-)Cat over general symmetric monoidal (∞, 1)-categories. We prove that Ad\'amek's construction of an initial algebra for (-)Cat does not terminate; more precisely, we show that Ad\'amake's construction of an initial algebra for the endofunctor (-)Cat<λ that sends a symmetric monoidal (∞, 1)-category V to the (∞, 1)-category of V-enriched categories with at most λ equivalence classes of objects terminates in precisely λ steps. We also prove that an initial algebra for the endofunctor (-)Cat exists nonetheless, and characterise it as the (∞, 1)-category consisting of those (∞, ∞)-categories that satisfy a weak finiteness property we call Noetherian.