  We study the semigroup of non-injective monoid endomorphisms of the semigroup
$\boldsymbol{B}_{\omega}^{\mathscr{F}}$ with a two-elements family
$\mathscr{F}$ of inductive nonempty subsets of $\omega$. We describe the
structure of elements of the semigroup
$\boldsymbol{End}^*_0(\boldsymbol{B}_{\omega}^{\mathscr{F}})$ of non-injective
monoid endomorphisms of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$.
In particular we show that its subsemigroup
$\boldsymbol{End}^*(\boldsymbol{B}_{\omega}^{\mathscr{F}})$ of non-injective
non-annihilating monoid endomorphisms of the semigroup
$\boldsymbol{B}_{\omega}^{\mathscr{F}}$ is isomorphic to the direct product of
the two-element left-zero semigroup and the multiplicative semigroup of
positive integers and describe Green's relations on
$\boldsymbol{End}^*(\boldsymbol{B}_{\omega}^{\mathscr{F}})$.
