  In the paper we describe injective endomorphisms of the inverse semigroup
$\boldsymbol{B}_{\omega}^{\mathscr{F}}$, which is introduced in the paper [O.
Gutik and M. Mykhalenych, \emph{On some generalization of the bicyclic monoid},
Visnyk Lviv. Univ. Ser. Mech.-Mat. \textbf{90} (2020), 5--19 (in Ukrainian)],
in the case when the family $\mathscr{F}_n$ is generated by the set
$\{0,1,\ldots,n\}$. In particular we show that the semigroup of injective
endomorphisms of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ is
isomorphic to $(\omega,+)$. Also we describe the structure of the semigroup
$\mathfrak{End}(\mathscr{B}_{\lambda})$ of all endomorphisms of the semigroup
of $\lambda{\times}\lambda$-matrix units $\mathscr{B}_{\lambda}$.
