  Given a morphism of (small) groupoids with injective object map, we provide
sufficient and necessary conditions under which the induction and co-induction
functors between the categories of linear representations are naturally
isomorphic. A morphism with this property is termed a Frobenius morphism of
groupoids. As a consequence, an extension by a subgroupoid is Frobenius if and
only if each fibre of the (left or right) pull-back biset has finitely many
orbits. Our results extend and clarify the classical Frobenius reciprocity
formulae in the theory of finite groups, and characterize Frobenius extension
of algebras with enough orthogonal idempotents.
