satisfying the following condition: $$\forall~ x,y\in G~~\hbox{such that}~~(f(x),f(y))\in
G_{(2)}^{\prime}~~~\Rightarrow~~~(x,y)\in G_{(2)}\label{2.2}$$ will be called [**strong morphism**]{} or [**homomorphism of groupoids**]{}.

Let the symmetry groupoid $ {\cal SG}_{n}$ of the finite set\
$ X = \{ x_{1}, x_{2},\ldots, x_{n} \} $ and the multiplicative group $\{ +1, -1 \}$ ( regarded as groupoid over $\{ +1\}$ ). We define the map\
$~~~~~~~ sgn^{\sharp}: {\cal SG}_{n} \to \{ +1, -1
\},~ f\in {\cal SG}_{n} \longmapsto sgn^{\sharp}(f):= sgn(f)
$,\
where $ sgn(f)$ is the signature of the permutation $f$ of degree $ k = | D(f)| $.

We have that [*$~ sgn^{\sharp}: {\cal SG}_{n} \to \{ +1, -1
\}~$ is a groupoid morphism*]{}.

Indeed, let $ f,g \in G_{(2)} $, where $ G = {\cal SG}(A,X)$ such that $ D(f) = D(g):= A_{k}:=\{ x_{j_{1}},\ldots,
x_{j_{k}}\}\subseteq X, ~ 1\leq k \leq n.$ Then $ f $ and $ g $ are permutations of $ A_{k} $ and $ f\circ g $ is also a permutation of $ A_{k}$. It is clearly that the condition (i) from Definition 2.2 is verified. Also, it is well known that\
$sgn(f\circ g) = sgn(f)\cdot sgn(g)$. Hence $~ sgn^{\sharp}(m(f,g)) =
sgn^{\sharp}(f)\cdot sgn^{\sharp}(g).~$ Therefore the condition (ii) from Definition 2.2 holds.

