In Accessible Categories: The Foundations of Categorical Model Theory (Makkai, Paré), it is said (Introduction, p. 3):
One sign of the “rightness” of the notion [of locally presentable category] is that it is easy to characterize the categories of algebras of equational theories, the ones that the theory of monads is for, as locally presentable categories satisfying a simple additional property.
What is this additional property that Makkai and Paré are talking about?