The pullback theorem for relative monads

A fundamental result in the theory of monads is the characterisation of the category of algebras for a monad in terms of a pullback of the category of presheaves on the category of free algebras: intuitively, this expresses that every algebra is a colimit of free algebras. We establish an analogous result for enriched relative monads with dense roots, and explain how it generalises the nerve theorems for monads with arities and nervous monads. As an application, we derive sufficient conditions for the existence of algebraic colimits of relative monads. More generally, we show that the pullback theorem for relative monads holds in any exact virtual equipment. In doing so, we are led to study the relationship between a j-relative monad T and its associated loose-monad E (j, T), and consequently show that the opalgebra object and the algebra object for T may be constructed from certain double categorical limits and colimits associated to E (j, T).