  We study relative amenability and amenability of a right coideal
$\widetilde{N}_P\subseteq \ell^\infty(\mathbb{G})$ of a discrete quantum group
in terms of its group-like projection $P$. We establish a notion of a $P$-left
invariant state and use it to characterize relative amenability. We also
develop a notion of coamenability of a compact quasi-subgroup
$N_\omega\subseteq L^\infty(\widehat{\mathbb{G}})$ that generalizes
coamenability of a quotient as defined by Kalantar, Kasprzak, Skalski, and
Vergnioux, where $\widehat{\mathbb{G}}$ is the compact dual of $\mathbb{G}$. In
particular, we establish that the coamenable compact quasi-subgroups of
$\widehat{\mathbb{G}}$ are in one-to-one correspondence with the idempotent
states on the reduced $C^*$-algebra $C_r(\widehat{\mathbb{G}})$. We use this
work to obtain results for the duality between relative amenability and
amenability of coideals in $\ell^\infty(\mathbb{G})$ and coamenability of their
codual coideals in $L^\infty(\widehat{\mathbb{G}})$, making progress towards a
question of Kalantar et al{.}.
