The n-Category Café

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.