### Monads and Lawvere Theories

#### Posted by John Baez

*guest post by Jade Master*

I have a question about the relationship between Lawvere theories and monads.

Every morphism of Lawvere theories $f \colon T \to T'$ induces a morphism of monads $M_f \colon M_T \Rightarrow M_{T^'}$ which can be calculated by using the universal property of the coend formula for $M_T$. (This can be found in Hyland and Power’s paper Lawvere theories and monads.)

On the other hand $f \colon T \to T'$ gives a functor $f^\ast \colon Mod(T') \to Mod(T)$ given by precomposition with $f$. Because everything is nice enough, $f^\ast$ always has a left adjoint $f_\ast \colon Mod(T) \to Mod(T')$. (Details of this can be found in *Toposes, Triples and Theories*.)

My question is the following:

What relationship is there between the left adjoint $f_\ast \colon Mod(T) \to Mod(T')$ and the morphism of monads computed using coends $M_f \colon M_T \Rightarrow M_{T^'}$?

In the examples I can think of the components of $M_f$ are given by the unit of the adjunction between $f^\ast$ and $f_\ast$ but I cannot find a reference explaining this. It doesn’t seem to be in *Toposes, Triples, and Theories*.

## Re: Monads and Lawvere Theories

I don’t have the time or brainpower to work it out right now, but the first thing I would try is to mess around with the calculus of mates for an arbitrary morphism of monads whose induced functor on algebras has a left adjoint, not assuming that it comes from a Lawvere theory.