A family of groups as a monoidal category 1.Context
My lecture notes present the following example of a monoidal category:

Let $G:=(G_n)_{n\in \mathbb {N_0}}$ be a family of groups with $G_0$ the trivial group with one element. We define a category $C_G$ with:

*

*$Obj(C_G)= \mathbb {N_0}$

*$Hom(m,n)=  \begin{cases} G_m & m=n \\
                       \emptyset & m \neq n \\
                    \end{cases}$

*Composition of two morphisms $f,h: m \rightarrow m$ is given by the (associative) multiplication in the respective group $G_m$. The identity morphism for each $m\in  \mathbb {N_0}$ is given by the neutral element of $G_m$.

Now suppose that for any $m,n,k \in  \mathbb {N_0} $ we have a group homomorphism $$\rho_{m,n}: G_m \times G_n \rightarrow G_{m+n}$$ such that $$\rho_{m+n,k} \circ (\rho_{m,n} \times id_{G_k})= \rho_{m,n+k} \circ (id_{G_m} \times \rho_{n,k}).$$ Then we can equip the category $C_G$ with a strict monoidal structure in the following (natural) way:
Define a functor $\otimes: C_G \times C_G \rightarrow C_G$ with $m \otimes n := m + n$ and $f \otimes h := \rho_{m,n}(f,h)$ for $f \in G_m, h \in G_n$. The monoidal unit is given by $0 \in \mathbb N_0$.

The essential property of this category seems to be, that it allows two types of operations (on certain families of groups): The standard multiplication within a group ("vertical composition") and a operation between elements of different groups ("horizontal composition").
By the way, unless I am overlooking something, a similar construction to the one above should work for certain families of associative monoids as well, i.e. we didn't use the inverse of a group element in the construction.
2. Questions

*

*Does the above category (its construction) have a name?

*My lecture notes mention the category in passing. It seems like a cute example of a monoidal category, that illustrates the versatility of the definition. Is there more to it? Is the category, in particular considered as a monoidal category, used anywhere? That is, are there any results where it is of interest?

*What (interesting or enlightening) examples of the above category exist?
An example presented in my lecture notes is $G:=(B_n)_{n\in \mathbb {N_0}}$ where $B_n$ denotes the braid group with $n$ strands. The group homomorphism $\rho_{m,n}$ is given by $\rho_{m,n} (\sigma_i, \sigma_j)= \sigma_i \sigma_{m+j}$ for each $m,n \in \mathbb {N_0} $, i.e. by the juxtaposition of braids. Here, $\sigma_i$ denotes one of the respective canonical generators of the braid group. Obviously, a similiar construction works for  the family of symmetric groups $G:=(S_n)_{n\in \mathbb {N_0}}$.
The family $G:=(C_n)_{n\in \mathbb {N_0}}$ of cyclic groups (or dihedral groups, for that matter) does not seem to work. I wouldn't know what the group homomorphisms $\rho_{m,n}$ should be. (One might modify the above construction so that $\rho_{m,n}: G_m \times G_n \rightarrow G_{lcm(m,n)}$. However, I would like to stick to above construction).

 A: This is not a full answer, but too long for a comment.
I don't know if that category has a name.
A place where it's used is the following : take $G_n = GL_n(F)$, for a field $F$. The morphisms $GL_n(F)\times GL_m(F)\to GL_{n+m}(F)$ are block sums; that is, if you have an automorphism $f:F^n\to F^n$, and $g:F^m\to F^m$, then you get $f\oplus g : F^{n+m}\to F^{n+m}$.
The category you get is often denoted $\coprod_n BGL_n(F)$ (because the one-object category associated to a group $G$ can be denote $BG$), and if you take its geometric realization, you get a nice space (the space associated to $BGL_n(F)$ has $\pi_1 = GL_n(F)$ and not other homotopy group).
This space is very closely related to the algebraic $K$-theory of $F$, and actually the monoidal structure on your category is a way to define an additive structure on the $K$-theory at the level of spaces.
(there are variants when you replace $F$ with an arbitrary ring)
The case of the family of symmetric groups $(S_n)$ is related to stable homotopy theory (specifically to the sphere spectrum and therefore the stable homotopy groups of spheres), and allegedly to the $K$-theory of "the field with one element"
A: Not a full answer either but here is what I can make out of it:
I don't know if this construction has a name, neither if it is used in any example, but I can provide you with some intuition about that:
First we need to start with the "delooping" of a monoid: this is kind of standard in category theory. The general takeaway is that monoids really are the same as categories with a single object. I won't go too much in details about this as it is really the same as what you have presented just the notations: given a monoid $M$, you can define the category $\mathbf{B}M$, which has a single object $\ast$ and the morphisms are given by $\mathbf{B}M(\ast,\ast) = M$. The composition of morphisms is given by the composition in $M$. In case you have not encountered this already, you can prove that $\mathbf{B}$ is an equivalence of categories between the category of monoids and the category of categories with a single object, and this has super slick generalizations to higher categories, but that's kind of a (very interesting) rabbit hole (see https://ncatlab.org/nlab/show/k-tuply+monoidal+n-category if you want more!).
Now what you define (without considering the monoidal at first) is like the delooping of monoid, except that you do it for a family of monoids. Luckily, these deloopings don't talk to each other (you have $\operatorname{Hom}(m,n)=\emptyset$ is $m\neq n$). So we can express it using the sum of the delooping. The sum of two categories $C$ and $D$, that I denote $C+D$, is the category whose objects is the disjoint union of the objects of $C$ and the object of $D$, and the morphisms are given by the moprhisms in $C$ and the morphisms in $D$, and there is no extra morphisms in between pairs of objects that come from a mix of $C$ and $D$. It is the categorical sum inside the category of categories, and you can picture it as haveing $C$ and $D$ side by side, but not interacting. Now, given a family of monoids $M_0,M_1,M_2,\ldots$, you can define the category you consider as the infinite sum $C_M = {\large+}_{i=0}^\infty \mathbf{B}M_i$. (Actually you would need to define this $\large{+}$ operation, but that's similar to the case where there are only 2 monoids).
Another way of picturing this is that $C_M$ is that it is a category that has an countable number of connected components, such that the connected components are exactly given by the list $M_0,M_1,M_2,\ldots$ An extra image you can adopt is that a category is kind of a big collection of monoids that interact with each other is a nice way, but here you just have monoids that forgot to interact with each other. I am giving as much images as I can, as maybe some of them will be more intuitive, but up until now I have only said one thing.
Now we can take a look at the additional condition, that there is a family of homomorphisms satisfying the identities you have mentioned. Then you can construct a monoidal structure on $C_M$, as you mentioned. But you can alo go the other way and check that a monoidal structure on $C_M$ is exactly the same as a family of homomorphisms satisfying your condition gives back a family of homomorphisms. Simply define $\rho_{m,n}(f,g) = f\otimes g$, and this family satisfies the equations you asked for. So in the end, a family of morphisms such as the one you asked for really is a synonymous to a monoidal structure on $C_M$. Now my opinion about this is that being a monoidal structure on $C_M$ is a much simpler and natural condition, that tells you everything about a family monoids equipped with homomorphisms that interact in a nice way in one go. So I would argue that this is the more primitive notion and that, if you ever encounter such families of morphisms, you should really understand them as a monoidal structure on a category $C_M$.
That being said, I don't know of any example of such structure, so I trust the example to be relevant!
