over $ {\cal
G}al({\cal E}/K)_{0}$, called the [*Galois groupoid associated to $ {\cal E}$*]{}. The isotropy group at $ u= Id_{E_{i}}$ is the Galois group $ Gal(E_{i}/K)$.

([@cdw87]) By **morpfism of groupoids** or [**groupoid morphism**]{} between the groupoids $(G ,\alpha ,\beta , m ,\iota
,G_0)$ and $(G ^{\prime },\alpha ^{\prime },\beta ^{\prime }, m
^{\prime },\iota ^{\prime} , G_{0}^{\prime})$, we mean a map $~f:G
\rightarrow G ^{\prime } $ which verifies the following conditions:

[*(i)*]{} $~~~~~\forall~ (x,y)\in G_{(2)}~~\Longrightarrow~~ (
f(x), f(y) )\in G_{(2)}^{\prime};~$

[*(ii)*]{}$~~~~~f( m(x,y)) = m^{\prime}(f(x),f(y)), ~ \forall ~
(x,y)\in G_{(2)}.$

[*If $ f : G \longrightarrow G ^{\prime}$ is a morpfism of groupoids, then: $$\hbox{{\it(a)}}~~~ f\left( u\right)\in G_{0}^{\prime },~~~ \forall~ u\in
G_{0};~~~~~ \hbox{{\it(b)}}~~ f\left( x^{-1}\right) =\left(
f\left( x\right) \right) ^{-1},~  \forall~ x\in G.$$*]{}

From Proposition 2.3(a) follows that a groupoid morphism $ f : G
\to G^{\prime} $ induces a map $ f_{0} : G_{0} \to G_{0}^{\prime}
$ taking $ f_{0}(u): = f(u), ~(\forall) u\in G_{0}$, i.e. the map $ f_{0}$ is the restriction of $ f$ to $ G_{0}.$ We say that $(f,f_{0}) : (G,G_{0}) \rightarrow ( G^{\prime}, G_{0}^{\prime} )
$ is a morphism of groupoids.

If $~G_{0} = G_{0}^{\prime}~$ and $~f_{0} = Id_{G_{0}},$ we say that 