When Jon Beck considers in his paper Distributive laws the paradigmatic distributive law of $S$ over $T$, where $S$ is the free monoid monad on $\mathbf{Set}$ and T is the free abelian group monad on $\mathbf{Set}$, he denotes the forgetful funtor from $\mathbf{Set}^{TS}=\mathbf{Ring1}$, the category of rings with unit, to $\mathbf{Set}^T=\mathbf{Ab}$, the category of abelian groups, as $\mathrm{Hom}_T(F^{TS},-)$, where $F^{TS}$ is the free ring-with-unit functor $\mathbf{Set}\rightarrow\mathbf{Set}^{TS}$. Why does he denote that functor like that?

He denotes the left adjoint of $\mathrm{Hom}_T(F^{TS},-)$ as $(-)\otimes_TF^{TS}$, and says that this is justifiable, as implying that it is because the forgetful functor $\mathbf{Ring1}\rightarrow\mathbf{Ab}$ can be denoted as shown above.


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