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

Homotopy theories of $(\infty, \infty)$-categories as universal fixed points with respect to enrichment