Let $F$ be a local field of characteristic zero and $G$ be a classical group over $F$.

Let $P=MN$ be a parabolic subgroup of $G$ and $\pi$ a irreducible smooth representation of $M$.

Let $\sigma$ be an irreducible constituent of normalized parabolic induction $\operatorname{Ind}_{M}^G(\pi)$. Then I am wondering whether the normalized Jacquet module $J_{N}(\sigma)$ has $\pi$ as a quotient? Some paper argues in this way without proof but I don't know the reason exactly. Why does it hold?

There is one more question. What is the difference of irreducible subquotient and irreducible consitituent of a module? I guess the people use the later when the given module is of finite length and the former is used in more general situations. Am I right?

Thank you very much!