If $X$ and $Y$ are smooth projective varieties, $p: X \times Y \to X$ and $q: X \times Y \to Y$ are the projections, and $\mathcal{P}$ is an object in $D^b(X \times Y)$, then the Fourier--Mukai transform associated to $\mathcal{P}$ is a functor $\Phi\_\mathcal{P}: D^b(X) \to D^b(Y)$ that sends $\mathcal{E}^\bullet$ to $q\_\*(p^\*\mathcal{E}^\bullet \otimes \mathcal{P})$ (where pushforward, pullback, and tensor are the derived versions).

It is a fact that if we define $\mathcal{P}\_L := \mathcal{P}^\vee \otimes q^\*\omega\_Y[\dim Y]$ and $\mathcal{P}\_R := \mathcal{P}^\vee \otimes p^\*\omega\_X[\dim X]$ (where $-^\vee$ is the derived dual), then $\Phi\_{\mathcal{P}\_L}$ is left-adjoint to $\Phi\_{\mathcal{P}}$ and $\Phi\_{\mathcal{P}\_R}$ is right-adjoint to $\Phi\_{\mathcal{P}}$. (This is Proposition 5.9 in Huybrechts's book on Fourier--Mukai transforms, and he credits it to Mukai.)

Something that pops out to me is that the left-adjoint and the right-adjoint to $\Phi\_\mathcal{P}$ are pretty similar, and that in fact they're the same if $X$ and $Y$ are Calabi--Yau and have the same dimension. My general impression is that it's a very special property for $(F, G)$ and $(G, F)$ to both be adjoint pairs (maybe it's the "next best thing" to $F$ and $G$ being inverse to one another?).

My question is:

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> Is this formal property that I mention in the third paragraph useful? What information about $D^b(X)$ and $D^b(Y)$ does it buy you, especially for special cases of $X$ and $Y$?
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I'm not an algebraic geometer, so obvious-seeming answers are appreciated as much as non-obvious-seeming answers. The reason I'm asking is because I'm studying the symplectic version of the Fourier--Mukai transform (see [this paper](http://math.mit.edu/~katrin/papers/ainfty.pdf)) and it seems to exhibit a similar formal property.