  Let $G_n$ be an inner form of a general linear group over a non-Archimedean
field. We fix an arbitrary irreducible representation $\sigma$ of $G_n$.
Lapid-M\'inguez give a combinatorial criteria for the irreducibility of
parabolic induction when the inducing data is of the form $\pi \boxtimes
\sigma$ when $\pi$ is a segment representation. We show that their criteria can
be used to define a full subcategory of the category of smooth representation
of some $G_m$, on which the parabolic induction functor $\tau \mapsto \tau
\times \sigma$ is fully-faithful. A key ingredient of our proof for the
fully-faithfulness is constructions of indecomposable representations of length
2.
  Such result for a special situation has been previously applied in proving
the local non-tempered Gan-Gross-Prasad conjecture for non-Archimedean general
linear groups. In this article, we apply the fully-faithful result to prove a
certain big derivative arising from Jacquet functor satisfies the property that
its socle is irreducible and has multiplicity one in the Jordan-H\"older
sequence of the big derivative.
