Hi, the reference you are searching comes under the title "adjoint triangle theorem" which is due to Eduardo Dubuc (cf. the nLab entry). The left adjoint f_* is uniquely determined by the fact that M_T and M_T' are "nice" monads. The explicit (known) formulas for this left adjoint imply your observation. All the best, Clemens. Le 2018-12-22 18:45, Jade Master a ??crit??:
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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