  We describe injective monoid endomorphisms of the semigroup
$\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$ with a three element family
$\mathscr{F}^3$ of inductive nonempty subsets of $\omega$. Also, we show that
the monoid $\boldsymbol{End}_*^1(\boldsymbol{B}_{\omega}^{\mathscr{F}})$ of all
injective endomorphisms of the semigroup
$\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$ is isomorphic to the multiplicative
semigroup of positive integers.
