By (1) and Lemma \ref{lemma-subcategory-left-resolution} for any bounded above complex $K^\bullet$ there exists a quasi-isomorphism $P^\bullet \to K^\bullet$ with $P^\bullet$ bounded above and $P^n \in \mathcal{P}$ for all $n$. Suppose that $s : P^\bullet \to (P')^\bullet$ is a quasi-isomorphism of bounded above complexes consisting of objects of $\mathcal{P}$. Then $F(P^\bullet) \to F((P')^\bullet)$ is a quasi-isomorphism because $F(C(s)^\bullet)$ is acyclic by assumption (2). This already shows that $LF$ is defined on $D^{-}(\mathcal{A})$ and that a bounded above complex consisting of objects of $\mathcal{P}$ computes $LF$, see Lemma \ref{lemma-find-existence-computes}. \medskip\noindent Next, let $K^\bullet$ be an arbitrary complex of $\mathcal{A}$. Choose a diagram $$ \xymatrix{ P_1^\bullet \ar[d] \ar[r] & P_2^\bullet \ar[d] \ar[r] & \ldots \\ \tau_{\leq 1}K^\bullet \ar[r] & \tau_{\leq 2}K^\bullet \ar[r] & \ldots } $$ as in Lemma \ref{lemma-special-direct-system}. Note that the map $\colim P_n^\bullet \to K^\bullet$ is a quasi-isomorphism because colimits over $\mathbf{N}$ in $\mathcal{A}$ are exact and $H^i(P_n^\bullet) = H^i(K^\bullet)$ for $n > i$. We claim that $$ F(\colim P_n^\bullet) = \colim F(P_n^\bullet) $$ (termwise colimits) is $LF(K^\bullet)$, i.e., that $\colim P_n^\bullet$ computes $LF$. To see this, by Lemma \ref{lemma-find-existence-computes}, it suffices to prove the following claim. Suppose that $$ \colim Q_n^\bullet = Q^\bullet \xrightarrow{\ \alpha\ } P^\bullet = \colim P_n^\bullet $$ is a quasi-isomorphism of complexes, such that each $P_n^\bullet$, $Q_n^\bullet$ is a bounded above complex whose terms are in $\mathcal{P}$ and the maps $P_n^\bullet \to \tau_{\leq n}P^\bullet$ and $Q_n^\bullet \to \tau_{\leq n}Q^\bullet$ are quasi-isomorphisms. Claim: $F(\alpha)$ is a quasi-isomorphism. \medskip\noindent The problem is that we do not assume that $\alpha$ is given as a colimit of maps between the complexes $P_n^\bullet$ and $Q_n^\bullet$. However, for each $n$ we know that the solid arrows in the diagram $$ \xymatrix{ & R^\bullet \ar@{..>}[d] \\ P_n^\bullet \ar[d] & L^\bullet \ar@{..>}[l] \ar@{..>}[r] & Q_n^\bullet \ar[d] \\ \tau_{\leq n}P^\bullet \ar[rr]^{\tau_{\leq n}\alpha} & & \tau_{\leq n}Q^\bullet } $$ are quasi-isomorphisms. Because quasi-isomorphisms form a multiplicative system in $K(\mathcal{A})$ (see Lemma \ref{lemma-acyclic}) we can find a quasi-isomorphism $L^\bullet \to P_n^\bullet$ and map of complexes $L^\bullet \to Q_n^\bullet$ such that the diagram above commutes up to homotopy. Then $\tau_{\leq n}L^\bullet \to L^\bullet$ is a quasi-isomorphism. Hence (by the first part of the proof) we can find a bounded above complex $R^\bullet$ whose terms are in $\mathcal{P}$ and a quasi-isomorphism $R^\bullet \to L^\bullet$ (as indicated in the diagram). Using the result of the first paragraph of the proof we see that $F(R^\bullet) \to F(P_n^\bullet)$ and $F(R^\bullet) \to F(Q_n^\bullet)$ are quasi-isomorphisms. Thus we obtain a isomorphisms $H^i(F(P_n^\bullet)) \to H^i(F(Q_n^\bullet))$ fitting into the commutative diagram $$ \xymatrix{ H^i(F(P_n^\bullet)) \ar[r] \ar[d] & H^i(F(Q_n^\bullet)) \ar[d] \\ H^i(F(P^\bullet)) \ar[r] & H^i(F(Q^\bullet)) } $$ The exact same argument shows that these maps are also compatible as $n$ varies. Since by (4) and (5) we have $$ H^i(F(P^\bullet)) = H^i(F(\colim P_n^\bullet)) = H^i(\colim F(P_n^\bullet)) = \colim H^i(F(P_n^\bullet)) $$ and similarly for $Q^\bullet$ we conclude that $H^i(\alpha) : H^i(F(P^\bullet) \to H^i(F(Q^\bullet)$ is an isomorphism and the claim follows.