Recall that an action of a groupoid G ⇉ ⇉ M on a set P along a function μ μ : P ⇉ ⇉ M is defined via a collection of maps μ μ − − 1 ( x ) → → μ μ − − 1 ( y ) , p ↦ ↦ g ⋅ ⋅ p for each morphism g ∈ ∈ G between x , y ∈ ∈ M . Accordingly, an action of a Lie groupoid G ⇉ ⇉ M on a manifold P along a smooth map μ μ : P ⇉ ⇉ M consists of a groupoid action where the maps μ μ − − 1 ( x ) → → μ μ − − 1 ( y ) are smooth. Of course, for every x ∈ ∈ M there is an induced smooth action of the isotropy group G x on the fibre μ μ − − 1 ( x ) .