$~f : G \to G^{\prime}~$ is a $ G_{0}~$- [*morphism*]{} of groupoids over $ G_{0}$.

A groupoid morphism $~(f,f_{0})~$ is said to be [*isomorphism of groupoids*]{} or [*groupoid isomorphism*]{}, if $ f $ and $ f_{0}$ are bijective maps.

([@ivan02])

*Let $(G ,\alpha ,\beta, m ,\iota ,G_0)$ and $(G^{\prime},\alpha^{\prime }, \beta^{\prime}, m^{\prime
},\iota^{\prime },G_{0}^{\prime})$ be two groupoids. The pair $~(f,f_{0}) : (G, G_{0})\longrightarrow (G^{\prime},
G_{0}^{\prime})~$ where $ f:G \longrightarrow G^{\prime}$ and $ ~
f_{0} : G_{0}\longrightarrow G_{0}^{\prime},$ is a groupoid morphism if and only if the following conditions are verified:*

[*(i)*]{} $~~~~~\alpha^{\prime} \circ f = f_{0}\circ \alpha \quad
\hbox{and}\quad  \beta^{\prime} \circ f = f_{0}\circ \beta ;$

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

Applying Propositions 2.3 and 3.4 we can conclude that a groupoid morphism $~(f,f_{0}) : (G,
G_{0})\longrightarrow (G^{\prime}, G_{0}^{\prime})~$ is linked with the structure functions by the relations : $$\alpha^{\prime} \circ f = f_{0} \circ \alpha ,~~ \beta^{\prime}
\circ f = f_{0} \circ \beta ,~~ m^{\prime} \circ (f \times f) =
f\circ m,~~ \iota^{\prime} \circ f = f \circ \iota \label{2.1}$$ where $~(f\times f)(x,y):=(f(x),f(y)),~ \forall~ x,y\in G\times
G.$

([@dumoiv]) A groupoid morphism $~( f,
f_{0} ):(G, G_{0})\longrightarrow (G^{\prime},G_{0}^{\prime})~$ 