Given any symmetric monoidal category $\mathbf{C}$, $\mathrm{Mon}(\mathbf{C})$ (the category of monoids, not necessarily commutative, in $\mathbf{C}$) is also a symmetric monoidal category.
Now, is there a known characterization of which symmetric monoidal categories could be realized as $\mathrm{Mon}(\mathbf{C})$ for some symmetric monoidal category $\mathbf{C}$?
Call a symmetric monoidal category $\mathbf{D}$ "monoid-realizable" if $\mathbf{D}$ is symmetric monoidally equivalent to $\mathrm{Mon}(\mathbf{C})$ for some symmetric monoidal category $\mathbf{C}$.
Then, monoid-realizable symmetric monoidal categories $\mathbf{D}$ must satisfy the following four conditions:
- The unit object $I$ in $\mathbf{D}$ is initial. (Needed to define the two morphisms in the next condition below)
- For any two objects $X$ and $Y$ of $\mathbf{D}$, the morphisms $i_{X,Y}:X \cong X \otimes I \to X \otimes Y$ and $j_{X,Y}:Y \cong I \otimes Y \to X \otimes Y$ are "jointly epic", i.e., if $f,g:X \otimes Y \to Z$ are two morphisms in $\mathbf{D}$ for which $f \circ i_{X,Y}=g \circ i_{X,Y}$ and $f \circ j_{X,Y}=g \circ j_{X,Y}$, then $f=g$.
- For any object $X$ of $\mathbf{D}$, $i_{X,X}$ and $j_{X,X}$ (see the second condition above for their definitions) are monic (this also implies that $i_{X,Y}$ and $j_{Y,X}$ are monic whenever there is a morphism $Y \to X$).
- For any object $X$ of $\mathbf{D}$, if the symmetry automorphism of $X \otimes X$ is the identity, then $X$ has a (commutative) monoid structure.
The second condition above is sufficient for the "Eckmann-Hilton argument" to apply. In particular, every object of $\mathbf{D}$ has at most one monoid structure, every monoid in $\mathbf{D}$ is commutative, and the forgetful functor $\mathrm{Mon}(\mathbf{D}) \to \mathbf{D}$ is full(y faithful).
Note that the first three conditions are not sufficient for a symmetric monoidal category $\mathbf{D}$ to be monoid-realizable. Indeed, if $\mathbf{D}$ is a preorder, then it must already be a cocartesian monoidal category (just consider the symmetry automorphism of $X \otimes X$ for any monoid $X$ in $\mathbf{C}$). So, $(\mathbf{R}_{\ge 0},\le,+,0)$ satisfies the first three conditions but is not monoid-realizable.
Even the above four conditions may still not be sufficient to conclude that a symmetric monoidal category $\mathbf{D}$ is monoid-realizable. One may also require an "opposite monoid" endofunctor of $\mathbf{D}$ that is the identity on $\mathrm{Mon}(\mathbf{D})$ (identified with a full subcategory of $\mathbf{D}$ since the forgetful functor is fully faithful), but even that may not be sufficient, and more conditions may still be needed.
One might think that "jointly epic" should be strengthened to "jointly strong epic" in the sense that $i_{X,Y}$ and $j_{X,Y}$ are jointly orthogonal to all monomorphisms (i.e., if $m:Z \to Z'$ is any monomorphism and $f:X \otimes Y \to Z'$ is any morphism, then $f$ factors through $m$ as soon as $f \circ i_{X,Y}$ and $f \circ j_{X,Y}$ do), but unless kernel pairs exist in a symmetric monoidal category $\mathbf{C}$, the forgetful functor $\mathrm{Mon}(\mathbf{C}) \to \mathbf{C}$ is not guaranteed to preserve monomorphisms in general, and so $i_{X,Y}$ and $j_{X,Y}$ are only guaranteed to be jointly orthogonal to monomorphisms that are preserved by the forgetful functor.
A somewhat-related question asked on MathOverflow $11$ years ago is Iterating monoid categories.