  Distributive laws give a way of combining two algebraic structures expressed
as monads; in this paper we propose a theory of distributive laws for combining
algebraic structures expressed as Lawvere theories. We propose four approaches,
involving profunctors, monoidal profunctors, an extension of the free
finite-product category 2-monad from Cat to Prof, and factorisation systems
respectively. We exhibit comparison functors between CAT and each of these new
frameworks to show that the distributive laws between the Lawvere theories
correspond in a suitable way to distributive laws between their associated
finitary monads. The different but equivalent formulations then provide,
between them, a framework conducive to generalisation, but also an explicit
description of the composite theories arising from distributive laws.
