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Since perfect complexes are dualizable, for every perfect complex $P$ and any complex $Q$ we have $$\mathrm{hom}(P,Q)\cong \mathrm{hom}(P,1)\otimes Q\,.$$ Moreover $\mathrm{hom}(P,1)$ is perfect too (since for qcqs schemes perfect=dualizable). In particular, by taking $Q=\omega$ the dualizing sheaf this shows that the image of the duality $D(-):=\mathrm{hom}(-,\omega)$ is exactly $\mathrm{Perf}(X)\otimes\omega$.