Lemma 57.11.1. Let F : \mathcal{D} \to \mathcal{D}' be an exact functor of triangulated categories. Let S \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{D}) be a set of objects. Assume
F has both right and left adjoints,
for K \in \mathcal{D} if \mathop{\mathrm{Hom}}\nolimits (E, K[i]) = 0 for all E \in S and i \in \mathbf{Z} then K = 0,
for K \in \mathcal{D} if \mathop{\mathrm{Hom}}\nolimits (K, E[i]) = 0 for all E \in S and i \in \mathbf{Z} then K = 0,
the map \mathop{\mathrm{Hom}}\nolimits (E, E'[i]) \to \mathop{\mathrm{Hom}}\nolimits (F(E), F(E')[i]) induced by F is bijective for all E, E' \in S and i \in \mathbf{Z}.
Then F is fully faithful.
Proof.
Denote F_ r and F_ l the right and left adjoints of F. For E \in S choose a distinguished triangle
E \to F_ r(F(E)) \to C \to E[1]
where the first arrow is the unit of the adjunction. For E' \in S we have
\mathop{\mathrm{Hom}}\nolimits (E', F_ r(F(E))[i]) = \mathop{\mathrm{Hom}}\nolimits (F(E'), F(E)[i]) = \mathop{\mathrm{Hom}}\nolimits (E', E[i])
The last equality holds by assumption (4). Hence applying the homological functor \mathop{\mathrm{Hom}}\nolimits (E', -) (Derived Categories, Lemma 13.4.2) to the distinguished triangle above we conclude that \mathop{\mathrm{Hom}}\nolimits (E', C[i]) = 0 for all i \in \mathbf{Z} and E' \in S. By assumption (2) we conclude that C = 0 and E = F_ r(F(E)).
For K \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D}) choose a distinguished triangle
F_ l(F(K)) \to K \to C \to F_ l(F(K))[1]
where the first arrow is the counit of the adjunction. For E \in S we have
\mathop{\mathrm{Hom}}\nolimits (F_ l(F(K)), E[i]) = \mathop{\mathrm{Hom}}\nolimits (F(K), F(E)[i]) = \mathop{\mathrm{Hom}}\nolimits (K, F_ r(F(E))[i]) = \mathop{\mathrm{Hom}}\nolimits (K, E[i])
where the last equality holds by the result of the first paragraph. Thus we conclude as before that \mathop{\mathrm{Hom}}\nolimits (C, E[i]) = 0 for all E \in S and i \in \mathbf{Z}. Hence C = 0 by assumption (3). Thus F is fully faithful by Categories, Lemma 4.24.4.
\square
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