Corollary 6.3.17. Let \(F\colon C \to D\) be an exact functor between stable \(\infty \)-categories with t-structures. Assume \(F\) restricts to a functor \(F\colon C^{\heartsuit } \to D^{\heartsuit }\). Then this restriction is an exact functor of abelian categories.
Proof. Let \(0 \to X \hookrightarrow Y \twoheadrightarrow Z \to 0\) be a short exact sequence in \(C^{\heartsuit }\). By the previous corollary, the sequence \(X \to Y \to Z\) is exact in \(C\), hence induces an exact sequence \(F(X) \to F(Y) \to F(Z)\) in \(D\). Using the various parts of Proposition 6.3.15, we see:
- (i)
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By (2), the morphism \(F(Y) \to F(Z)\) is an epimorphism in \(D^{\heartsuit }\), since its fiber \(F(X)\) lies in \(D^{\heartsuit }\);
- (ii)
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By (3), the morphism \(F(X) \to F(Y)\) is a monomorphism in \(D^{\heartsuit }\), since its cofiber \(F(Z)\) lies in \(D^{\heartsuit }\);
- (iii)
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By (4), the exact sequence \(F(X) \to F(Y) \to F(Z)\) in \(D\) is also exact in \(D^{\heartsuit }\).
We conclude that the sequence \(0 \to F(X) \hookrightarrow F(Y) \twoheadrightarrow F(Z) \to 0\) is exact in \(D^{\heartsuit }\), as desired. □
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