  We study injective endomorphisms of the semigroup
$\boldsymbol{B}_{\omega}^{\mathscr{F}}$ with the two-elements family
$\mathscr{F}$ of inductive nonempty subsets of $\omega$. We describe the
elements of the semigroup
$\boldsymbol{End}^1_*(\boldsymbol{B}_{\omega}^{\mathscr{F}})$ of all injective
monoid endomorphisms of the monoid $\boldsymbol{B}_{\omega}^{\mathscr{F}}$, and
show that Green's relations $\mathscr{R}$, $\mathscr{L}$, $\mathscr{H}$,
$\mathscr{D}$, and $\mathscr{J}$ on
$\boldsymbol{End}^1_*(\boldsymbol{B}_{\omega}^{\mathscr{F}})$ coincide with the
relation of equality.
