**Proof.**
Let $A$ be an object of $\mathcal{A}$. By Lemma 13.16.1 we have $H^ i(RF(A[0]) = 0$ for $i < 0$. This proves (1).

Let $0 \to A \to B \to C \to 0$ be a short exact sequence of $\mathcal{A}$. By Lemma 13.12.1 we obtain a distinguished triangle $(A[0], B[0], C[0], a, b, c)$ in $D^{+}(\mathcal{A})$. From the long exact cohomology sequence (and the vanishing for $i < 0$ proved above) we deduce that $0 \to R^0F(A) \to R^0F(B) \to R^0F(C)$ is exact. Hence $R^0F$ is left exact. Of course this also proves that if $F \to R^0F$ is an isomorphism, then $F$ is left exact.

Assume $F$ is left exact. Recall that $RF(A[0])$ is the value of the essentially constant system $F(K^\bullet )$ for $s : A[0] \to K^\bullet $ quasi-isomorphisms. It follows that $R^0F(A)$ is the value of the essentially constant system $H^0(F(K^\bullet ))$ for $s : A[0] \to K^\bullet $ quasi-isomorphisms, see Categories, Lemma 4.22.8. But if $s : A[0] \to K^\bullet $ is a quasi-isomorphism, then $A[0] \to \tau _{\geq 0}K^\bullet $ is a quasi-isomorphism. Hence in the category $A[0]/\text{Qis}^{+}(\mathcal{A})$ the quasi-isomorphisms $s : A[0] \to K^\bullet $ with $K^ n = 0$ for $n < 0$ are cofinal. It follows from Categories, Lemma 4.22.11 that we may restrict to such $s$. Moreover, for such an $s$ the sequence

\[ 0 \to A \to K^0 \to K^1 \]

is exact. Since $F$ is left exact we see that $0 \to F(A) \to F(K^0) \to F(K^1)$ is exact as well. It follows that $F(A) \to H^0(F(K^\bullet ))$ is an isomorphism and the system is actually constant with value $F(A)$. We conclude $R^0F = F$ as desired.
$\square$

## Comments (4)

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