Lemma 57.10.2. Let \mathcal{A} be an abelian category. Let \mathcal{D} be a triangulated category. Let F, F' : D^ b(\mathcal{A}) \longrightarrow \mathcal{D} be exact functors of triangulated categories. Assume
the functors F \circ i and F' \circ i are isomorphic where i : \mathcal{A} \to D^ b(\mathcal{A}) is the inclusion functor, and
for all X, Y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}) we have \mathop{\mathrm{Ext}}\nolimits ^ q_\mathcal {D}(F(X), F(Y)) = 0 for q < 0 (for example if F is fully faithful).
Then F and F' are siblings.
Proof.
Let K \in D^ b(\mathcal{A}). We will show F(K) is isomorphic to F'(K). We can represent K by a bounded complex A^\bullet of objects of \mathcal{A}. After replacing K by a translation we may assume A^ i = 0 for i > 0. Choose n \geq 0 such that A^{-i} = 0 for i > n. The objects
M_ i = (A^{-i} \to \ldots \to A^0)[-i],\quad i = 0, \ldots , n
form a Postnikov system in D^ b(\mathcal{A}) for the complex A^\bullet = A^{-n} \to \ldots \to A^0 in D^ b(\mathcal{A}). See Derived Categories, Example 13.41.2. Since both F and F' are exact functors of triangulated categories both
F(M_ i) \quad \text{and}\quad F'(M_ i)
form a Postnikov system in \mathcal{D} for the complex
F(A^{-n}) \to \ldots \to F(A^0) = F'(A^{-n}) \to \ldots \to F'(A^0)
Since all negative \mathop{\mathrm{Ext}}\nolimits s between these objects vanish by assumption we conclude by uniqueness of Postnikov systems (Derived Categories, Lemma 13.41.6) that F(K) = F(M_ n[n]) \cong F'(M_ n[n]) = F'(K).
\square
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