Let $X$ be an ordinary Gushel-Mukai fourfold and $Y$ its hyperplane section, which is a Gushel-Mukai threefold. I consider semi-orthogonal decompositions of $X$ and $Y$:
$D^b(X)=\langle\mathcal{K}u(X),\mathcal{O}_X,\mathcal{U}^{\vee},\mathcal{O}_X(H),\mathcal{U}^{\vee}(H)\rangle$ and $D^b(Y)=\langle\mathcal{K}u(Y),\mathcal{O}_Y,\mathcal{E}^{\vee}\rangle$, where $\mathcal{U}$ is tautological sub-bundle on GM-fourfold and $\mathcal{E}$ is tautological sub-bundle on GM-threefold. Now I define a functor from $\mathcal{K}u(Y)$ to $\mathcal{K}u(X)$ as follows. Let $j:Y\hookrightarrow X$, take $E\in\mathcal{K}u(Y)$, pushforward it to $X$, we get $j_*E\in D^b(X)$, then projection it to $\mathcal{K}u(X)$, we get $\mathrm{pr}_X(j_*E)\in\mathcal{K}u(X)$. Denote this functor by $F:E\mapsto\mathrm{pr}_X(j_*E)$. My question is does this functor have an adjoint functor from $\mathcal{K}u(X)$ to $\mathcal{K}u(Y)?$.
For my purpose, I need to compute $\mathrm{Ext}^1(\mathrm{pr}_X(j_*E),\mathrm{pr}_X(j_*E))$, but usually it is very complicated since after $\mathrm{pr}_X$, we get a very "big" complex, then it is hard to compute further. I was wondering if it is possible to use adjunction to reduce the computation to $\mathcal{K}u(Y)$, which is much simpler!
