I'm sorry for the error, I'm new to overflow. The link I meant is of course ncatlab.org/nlab/show/tensor+product+theory

The reason I'm asking is that, if it is; and the relationship between the resulting Lawvere theory and $T^{poly}_R$ is functorial, I can take this functor, compose it with $T^{lin}_\mathbb{Z} \otimes -$, and have a candidate for the right adjoint of the adjoint situation I was looking for. EDIT: With the tensor product I mean of course <a href="ncatlab.org/nlab/show/tensor+product+theory">the tensor product of algebraic theories</a>.

Thank you, that is indeed helpful. The way to make this precise is the fact that for two theories $T$ and $T'$ we have $\textbf{Mod}(T \otimes T', Set) \cong \textbf{Mod}(T, \textbf{Mod}(T', Set))$. Is the slice category of Commutative $R$-Algebras over $R$ again a category of models for a Lawvere-Theory?