\alpha, \beta,
\boxdot, \iota, V_{0}(u)=\{u\}) $ the [*isotropy vector groupoid*]{} at $u\in V_{0}$ of $V$, when one refers to the above structure given on it.

Let $ ( V_1, \alpha_1, \beta_1, V_{1,0} )$ and $ ( V_2, \alpha_2,
\beta_2, V_{2,0} )$ be two vector groupoids.

A groupoid morphism ( resp. groupoid homomorphism ) $
f:V_1\longrightarrow V_2 $ with property that $ f $ is a linear map, is called [**vector groupoid morphism**]{} ( resp. [**vector groupoid homomorphism**]{} ).

Let $(V,\alpha, \beta, \odot , \iota , V_{0})$ be a vector groupoid. We consider the pair vector groupoid $ ( V_{0}\times
V_{0}, {\widetilde}{\alpha}, {\widetilde}{\beta}, {\widetilde}{m}, {\widetilde}{\iota}, \Delta_{V_{0}}) $. Then\
[*the anchor map $(\alpha, \beta): V \to V_{0}\times
V_{0} $ is a homomorphism of vector groupoids between the vector groupoids $ V $ and $ V_{0}\times V_{0}$*]{}.

Indeed, if we denote $ (\alpha, \beta):=f $ and consider the elements $ x,y\in G $ such that $ ( f(x), f(y))\in (V_{0}\times
V_{0})_{(2)}$, then $ {\widetilde}{\beta}(f(x))= {\widetilde}{\alpha}(f(y)) $ and we have $ {\widetilde}{\beta}(\alpha(x), \beta(x))= {\widetilde}{\alpha}(\alpha(y),
\beta(y))~\Rightarrow ~ (\beta(x),
\beta(x))=(\alpha(y),\alpha(y))~\Rightarrow~ \beta(x))=\alpha(y)
$, i.e. $ (x,y)\in V_{(2)}. $ Therefore the condition (i) from Definition 2.2 holds.

For $(x,y)\in V_{(2)}$ we have

$ f(m(x,y))= f(xy)=(\alpha(xy),
\beta(xy))= (\alpha(x), \beta(y)) 