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^< that sends a symmetric monoidal (, 1) -category V to the (, 1) -category of V-enriched categories with at most equivalence classes of objects terminates in precisely 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.