Category of monoids isomorphic to coslice category

Question

Let $k$ be a commutative ring and $R$ a $k$-algebra. The category $R\text{-Bimod}$ of $R$-bimodules becomes a monoidal category with the tensor product of $R$-bimodules. Denote the category of $k$-modules by $k$-Mod. The category of monoid objects $\text{Mon}(R\text{-Bimod})$ is isomorphic to the undercategory $R/\text{Mon}(k\text{-Mod}).$ Similarly the category of commutative $k$-algebras is isomorphic to the coslice category of commutative rings under $k$. For another example where a category of monoids is isomorphic to a certain coslice category see Remark 5.5 here.

Question.
Is there an abstract nonsense proof of the fact that these categories of monoids are isomorphic coslice categories? Are they all instances of a more general result? I can verify that these categories are isomorphic, but I would prefer a conceptual explanation.

Answer 1

These are instances of a more general result. The following appears as Proposition 10.2 in Lucyshyn-Wright's Enriched algebraic theories and monads for a system of arities.

Proposition. Given a monoid $K$ in a monoidal category $\mathscr C$ with reflexive coequalizers that are preserved by $\otimes$ in each variable, there is an isomorphism: $$\text{Mon}(\text{Bimod}_{\mathscr C}(K)) \cong K/\text{Mon}(\mathscr C)$$

In fact, there is an even more general result, in which we generalise from monoids in a monoidal category to monads in a double category. This appears as Theorem 2.3.18 in Spivak–Schultz–Rupel's String diagrams for traced and compact categories are oriented 1-cobordisms, but I won't state it here since it requires some set up.

Unfortunately, I don't know of an entirely abstract proof of the statement, but it is fairly straightforward to verify at the level of generality of monoids in a monoidal category.