More on soundness in the enriched context

Working within enriched category theory, this paper extends the concept of soundness—originally introduced by Adámek, Borceux, Lack, and Rosický for ordinary categories—to the enriched setting. Specifically, it explores three main aspects: (1) the development of the theory of locally Φ-presentable V-categories for a sound class Φ, (2) the investigation of whether every Φ-accessible V-category is also Ψ-accessible for given sound classes Φ ⊆ Ψ, and (3) the introduction of a notion of Φ-ary equational theory, where the V-categories of models correspond to algebras for Φ-ary monads on V.