This question is motivated by the construction of the Kuznetsov component on a prime Fano threefold $X$ of index 1 (say genus $g \geq 6$, $g \neq 7, 9$):
$$
D^b(X) = \langle Ku(X), E, \mathcal{O}\_X \rangle
$$
where $E$ is the pullback of the rank 2 tautological subbundle on $Gr(2, g/2 + 2)$.


In [1], the authors construct explicit formulas for the gluing data of $E$ to $Ku(X)$ by considering $D := \langle \mathcal{O}\_X \rangle^\perp = \langle Ku(X), E \rangle$ along with inclusion functor $i : Ku(X) \hookrightarrow D$ and go on to state the left adjoint is $i^{\ast} = \mathbf{L}\_E$.


Now consider the inclusion $j : D \hookrightarrow D^b(X)$. Clearly $j \circ i$ is the inclusion of $Ku(X)$ into $D^b(X)$, so that the left adjoint $( j \circ i )^\ast$ is the usual projection functor defined for semi-orthogonal decompositions. What would the left adjoint $j^\ast$ be in this case?


[1] Jacovskis, Liu, Zhang. *Brill-Noether Theory for Kuznetsov Components and Refined Categorical Torelli Theorems for Index One Fano Threefolds*, 2022



Left adjoint for nested admissible categories
If $\mathcal{A} \subset \mathcal{T}$ is a right admissible subcategory and
$$
\mathcal{T} = \langle \mathcal{A}^\perp, \mathcal{A} \rangle
$$
is the corresponding semiorthogonal decomposition, the left mutation functor $\mathbf{L}\_{\mathcal{A}}$ is **defined** as the left adjoint functor of the embedding $\mathcal{A}^\perp \hookrightarrow \mathcal{T}$.


In particular, in your case $j^\* = \mathbf{L}\_{\mathcal{O}\_X}$.


