# Frobenius reciprocities

An adjunction of the form $$\mathrm{Hom}(A \otimes X, Y) \cong \mathrm{Hom}(X, A^* \otimes Y)$$ in a rigid monoidal category is sometimes called Frobenius reciprocity. Is there a result that unifies this adjunction with the familiar adjunction between the induction and the restriction of representations? What explains this coincidence in terminology?