  We show that both the $\infty$-category of $(\infty, \infty)$-categories with
inductively defined equivalences, and with coinductively defined equivalences,
satisfy universal properties with respect to weak enrichment in the sense of
Gepner and Haugseng. In particular, we prove that $(\infty, \infty)$-categories
with coinductive equivalences form a terminal object in the $\infty$-category
of fixed points for enrichment, and that $(\infty, \infty)$-categories with
inductive equivalences form an initial object in the subcategory of locally
presentable fixed points. To do so, we develop an analogue of Ad\'amek's
construction of free endofunctor algebras in the $\infty$-categorical setting.
We prove that $(\infty, \infty)$-categories with coinductive equivalences form
a terminal coalgebra with respect to weak enrichment, and $(\infty,
\infty)$-categories with inductive equivalences form an initial algebra with
respect to weak enrichment.
