Say you have some kind of "algebraic" category $A$ with a forgetful functor $U : A \to \mathbf{Set}$ which has a left adjoint $F : \mathbf{Set} \to A$. The natural transformations $U \to U$ can be interpreted as the terms of the algebraic theory. For example, if $A=\mathbf{CRing}$, these are just polynomials. Suppose we consider the map $\Phi : \mathrm{Nat}(id_A,id_A) \to \mathrm{Nat}(U,U)$ that simply takes a natural transformation $\eta : id_A \to id_A$ and performs horizontal composition with the identity natural transformation $U \to U$. What is the interpretation of the image of $\Phi$? That is, what are the terms that actually act as morphisms of $A$?

Why do $\mathbf{Ab}$ and related categories enjoy the property that $\Phi$ is surjective, i.e. every term acts as a morphism?

commutative: ncatlab.org/nlab/show/commutative+algebraic+theory $\endgroup$