Let \(p>3\) be a prime number, \(f\ge 1\) an integer. We consider a certain full subcategory \(\mathcal {C}\) of the category of smooth admissible mod p representations of either \({{\,\textrm{GL}\,}}_2{\textbf{Q}}_{p^f}\) or of the group of units of the quaternion algebra over \({\textbf{Q}}_{p^f}\) . This category was introduced in the context of the mod p Langlands program by [1] in the \({{\,\textrm{GL}\,}}_2\) -case and by [2] in the quaternion case. We prove that whether a smooth admissible mod p representation \(\pi \) (with central character) belongs to \(\mathcal {C}\) is completely determined by the restriction of \(\pi \) to an arbitrarily small open subgroup.
