Let $\mathscr{T}$ be a triangulated category, and $\mathscr{A}$ be a right admissible subcategory, which means that $i_{\mathscr{A}} : \mathscr{A} \rightarrow \mathscr{T}$ has a right adjoint $i_{\mathscr{A}}^R$. Let $\mathscr{B}$ another subcategory of $\mathscr{T}$ (not necessarily right/left admissible). Consider the subcategory $\mathscr{A} \cap \mathscr{B} \subset \mathscr{B}$. It's natural to ask whether this subcategory is still right admissible. The first thing one would try is to consider the right adjoint $i_{\mathscr{A}}^R$ and check whether it sends $\mathscr{A} \cap \mathscr{B}$ to $\mathscr{B}$. If this is the case, we get right admissibility. Let's assume however that this is not the case, i.e. there exists an object $E \in \mathscr{A} \cap \mathscr{B}$ such that $i_{\mathscr{A}}^R(E) \notin \mathscr{A} \cap \mathscr{B}$. Is it possible the there exists another functor which is right adjoint to $i_{\mathscr{A}} \vert_{\mathscr{A} \cap \mathscr{B}} : \mathscr{A} \cap \mathscr{B} \rightarrow \mathscr{B}$ and which is not the restriction of $i_{\mathscr{A}}^R$? I tried to apply the Yoneda lemma to prove that this is not possible, but the only case in which this works is the degenerate case in which $\mathscr{A} \cap \mathscr{B}$ weakly generates $\mathscr{B}$, i.e. if $\left( \mathscr{A} \cap \mathscr{B} \right)^{\perp} = 0$ in $\mathscr{B}$, which makes me think that maybe it is possible. Thank you in advance.