Lawvere theories and Monads

[Jade Master <jmast003@ucr.edu>]

I have a question about the relationship between Lawvere theories and
monads. Every morphism of Lawvere theories f: T ->T' induces a morphism of
monads M_f: M_T => M_T' which can be calculated by using the universal
property of the coend formula for M_T (this can be found in Hyland's
<https://www.irif.fr/~mellies/mpri/mpri-ens/articles/hyland-power-lawvere-theories-and-monads.pdf>
paper
on Lawvere theories and monads).

On the other hand f: T->T' gives a functor f* : Mod(T') -> Mod(T) given by
composition with f. Because everything is nice enough, f* always has a left
adjoint f_* : Mod(T) -> Mod(T') (details of this can be found here
<http://web.science.mq.edu.au/~street/MitchB.pdf> or in Toposes, Triples
and Theories).

My question is the following: What relationship is there between the
adjunction

  f_* \dashv f*: Mod(T) ->Mod(T')

and the morphism of monads computed using coends

M_f : M_T => M_T'?

In the examples I can think of the components of M_f are given by the unit
of the adjunction f_* \dashv f* but I cannot find a reference explaining
this. It doesn't seem to be in Toposes, Triples, and Theories.
<http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html>

Thank you,
Jade Master


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