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Apologies in advance if the following makes little to no sense, but here goes ..

Denote $m_G : Gtimes Gto G$ the multiplication of a group $G$. Does it make sense to think of the map $m_G$ as some kind of morphism in a (at the moment unspecified) category?

Even more, could we think of the family $m_G$ (indexed by the class of groups) as a natural transformation of some or other functors?

Let's call our mystery categories $mathcal A$ and $mathcal B$ with mystery functors $X,Y :mathcal Atomathcal B$ such that the natural transformation condition holds:
$$ m_HX(f) = Y(f)m_G $$
where $f$ is a morphism in $mathcal A$ and $G,H$ are groups.

For this to make sense, we need a category with objects $Gtimes G$ and $G$. So, we extend (?) the category of groups by $mathcal B_0 := mboxGrp_0 cup Gtimes G mid GinmboxGrp_0$. Morphisms in 'separate components' remain as they are in $mboxGrp$ and $mboxGrptimesmboxGrp$ respectively. There would be no morphisms of the form $Gto Htimes H$ and
$$mathcal B(Gtimes G,H) := varphi m_G mid varphi in mboxHom(G,H) $$
where for every $x,yin G$ $varphi m_G (x,y) := varphi (xy) = varphi (x)varphi (y) =: m_H(varphi,varphi)(x,y)$. The identities are $1_G$ or $(1_G,1_G)$ depending on the component and composition of the morphisms would happen naturally.

1. Is it guaranteed $(A,B) neq (A',B') implies mathcal B(A,B)capmathcal B(A',B') =emptyset, A,A',B,B'inmathcal B_0$?


2. Taking $mathcal A = mboxGrp$ with $X :mathcal Atomathcal B$ given such that $X(G) = Gtimes G$ and for every morphism $f:Gto H$, $X(f) = (f,f)$. Put $Y:mathcal Atomathcal B$ as the embedding, then we would have $m_G$ as a natural transformation $XRightarrow Y$.

I omit the routine checks here, for they aren't important for this discussion. I am interested in whether this idea of regarding families of operations as natural transformations is, call it, well-founded

Questionnaire.

1. Would such an approach be the only one? How else (if at all) would we regard the family $m_G$ as a natural transformation?


2. Is this a more general thing in universal algebra? Given a class of algebras with certain operations of various arities, could we regard every family of operations as a natural transformation? (For instance, inverse operation or unit element operation of groups)

Let $DeclareMathOperatorSetSetSet$ denote the category of sets and $DeclareMathOperatorGrpGrpGrp$ denote the category of groups.
Let $Upsilon:GrptoSet$ denote the forgetful functor and $Delta:SettoSet$ be the diagonal functor $Delta(X)=Xtimes X$.

Then you are looking for a natural transformation $mu:DeltacircUpsilontoUpsilon$.
If $X$ is a group, then $Upsilon(X)$ is its underlying set.
For each group $X$ let $mu_X:(DeltacircUpsilon)(X)toUpsilon(X)$ be its composition law.
If $f:Xto Y$ is a group homomorphism, then we have a commutative diagram$requireAMScd$:
beginCD
Upsilon(X)timesUpsilon(X)@=(DeltacircUpsilon)(X)@>mu_X>>Upsilon(X)\
@VUpsilon(f)timesUpsilon(f)VV@V(DeltacircUpsilon)(f)VV@VVUpsilon(f)V\
Upsilon(Y)timesUpsilon(Y)@=(DeltacircUpsilon)(Y)@>>mu_Y >Upsilon(Y)
endCD
which proves naturalness of $mu_X$.