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