Remark 56.15.3. If $F, F' : D^ b_{\textit{Coh}}(\mathcal{O}_ X) \to \mathcal{D}$ are siblings, $F$ is fully faithful, and $X$ is reduced and projective over $k$ then $F \cong F'$; this follows from Proposition 56.13.4 via the argument given in the proof of Theorem 56.16.3. However, in general we do not know whether siblings are isomorphic. Even in the situation of Lemma 56.15.2 it seems difficult to prove that the siblings $F$ and $F'$ are isomorphic functors. If $X$ is smooth and proper over $k$ and $F$ is fully faithful, then $F \cong F'$ as is shown in [Noah]. If you have a proof or a counter example in more general situations, please email stacks.project@gmail.com.

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