Forgive me if this is a well-known observation/result, but I'm quite new to graduate-level algebra and I was wondering if there are generalisations of the constructions I describe below.


It's straightforward to see that the functor "adjoin an indeterminate to a ring" is the unit of the adjunction $F : \mathbf{Ring} \to \mathbf{Ring}\_\* : U$, where $U: \mathbf{Ring}\_\* \to \mathbf{Ring}$ is the forgetful functor from the category of pointed rings to the category of rings. 


Along similar lines, the functor "disjoint-union an element to a set" is the unit of the adjunction $F : \mathbf{Set} \to \mathbf{Set}\_\* : U$, and if I'm not mistaken, the functor "coproduct with $\mathbb{Z}$" is the unit of the adjunction $F : \mathbf{Grp} \to \mathbf{Grp}\_\* : U$. Indeed, in general, it seems that if a category has a notion of "free object on one generator", a notion of "pointed" objects, and binary coproducts, then the forgetful functor from the category of pointed objects has a left adjoint, and the unit of the adjunction is the functor which takes objects to their coproduct with the free object on one generator. 


1. Does this construction generalise when binary coproducts don't exist?
2. What about when the required free object doesn't exist?
3. Is there a (even) more general way to describe this operation of "adjoining an indeterminate", e.g. when there's no notion of pointed objects?



Indeterminates and forgetful functors
In your examples, there is an evident underlying functor $U: C \to Set$ which allows you to talk about points as elements of the underlying set. So a pointed $C$-object would be an object of the comma category $1 \downarrow U$. There is an evident projection $\pi: 1 \downarrow U \to C$. A left adjoint to this functor would be the process of adjoining an indeterminate. 


A category $C$ equipped with an "underlying" functor $U$ to $Set$ is often called a *concrete category*, especially when $U$ is faithful (some authors drop this assumption). So concrete categories admit a notion of pointed object, and a notion of adjoining an indeterminate if the associated $\pi$ has a left adjoint $(-)[x]$. A priori this makes sense without the need to mention binary coproducts in $C$ or the free object on one element, although it seems to me instances of that would be somewhat artificial. (For example, if $C$ has an initial object $0$, then $0[x] = 1 \to Uc$ would exhibit $c$ as the free object on one element, precisely because $0[x]$ is initial in $1 \downarrow U$.) 


