Can the underlying set functor corresponding to an algebraic theory always be viewed as a model of that theory? Let $\mathsf{T}$ denote a Lawvere theory, and let $\mathbf{C}$ denote its category of models in $\mathbf{Set}$. Write $U : \mathbf{C} \rightarrow \mathbf{Set}$ for the underlying set functor. I think that $U$ can always be viewed as model of $\mathsf{T}$ in the functor category $[\mathbf{C},\mathbf{Set}].$ For example, suppose $\mathsf{T}$ has a binary operation $f : G \times G \rightarrow G$, where $G$ is the generic object of $\mathsf{T}$. Then there should be a corresponding natural transformation $\nu : U \times U \Rightarrow U$ given by writing $\nu_X = X(f)$ for all objects $X$ of $\mathbf{C}$. Hence $U$ becomes equipped with the operations of $\mathsf{T}$ in a natural way.
Is this right? If so, I'd appreciate some kind of a discussion of how this all works. Its currently very fuzzy to me.

If $f_G : G \times G \to G$ is a binary operation which belongs to $T$, then by definition homomorphisms preserve it, i.e. for every homomorphism $G \to H$ the diagram
$$\begin{array}{c} G \times G & \rightarrow & G \\ \downarrow && \downarrow \\ H \times H & \rightarrow & H \end{array}$$
commutes. In other words, we get a natural transformation $f : U \times U \to U$. Conversely, a natural transformation $U \times U \to U$ correponds by the Yoneda Lemma, since $U \times U$ is represented by the free $T$-algebra on two generators $x,y$, by an element of its underlying set, i.e. a derived binary operation of $T$. For example, in the Lawvere theory of groups, an example is $x^{-1} y x^2$.
