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?



Terms of an algebraic theory that act as homomorphisms
An algebraic theory in which every term is a homomorphism is called **commutative**: [ncatlab.org/nlab/show/commutative+algebraic+theory](http://ncatlab.org/nlab/show/commutative+algebraic+theory). I'm not sure that's *quite* what you were asking, because here "term" means "term in any number of variables", that is, natural transformation $U^n \to U$ for any $n$ (not just $1$). For instance, in the theory of abelian groups, $+: A \times A \to A$ is a homomorphism for any abelian group $A$. But commutative theories are certainly a useful class to consider, and a lot has been written about them. 


