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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]