Assume I have an abelian category $\mathcal C$ with enough projectives. Say the indecomposable projectives $P_i$ are indexed by $i\in I$ for some index set $I$.
Let $F\colon \mathcal C \to \mathcal C$ be a right exact functor of which I know the images $M_j = FP_j$ for all $j\in I$. Assume that I know projective resolutions $\cdots \to P_{i_{j,2}}\to P_{i_{j,1}} \to 0$ of all $M_j$. Now, for an object $N \in \mathcal C$ with projective resolution $\cdots\to P_{n_2} \to P_{n_1} \to 0$ of $N$, the image $\mathbf LFN$ under the left derived functor $\mathbf LF\colon \mathcal C \to D^+(\mathcal C)$ can be obtained by applying $F$ to the projective resolution $P_{n_\bullet}$ of $N$, which gives the complex $\cdots\to M_{n_2}\to M_{n_1}\to 0$ that should be quasi-isomorphic to $\mathbf LFN$. But since each $M_{n_k}$ is quasi-isomorphic to $P_{i_{n_k,\bullet}}$, shouldn't it be possible to determine $\mathbf LFN$ only from the resolutions $P_{i_{n_k,\bullet}}$ of the $M_{n_k}$? Of course, the structure
$$\begin{matrix}&& \vdots && \vdots\\&& \downarrow && \downarrow\\\cdots & \rightarrow & P_{i_{n_2,2}} & \rightarrow & P_{i_{n_2,1}}\\&& \downarrow && \downarrow\\\cdots & \rightarrow & P_{i_{n_2,1}} & \rightarrow & P_{i_{n_1,1}}\end{matrix} \tag{$\dagger$}$$
is no double complex; the columns are exact since they are resolutions of the $M_{n_k}$'s, but horizontally, the structure is only exact up to chain homotopy.
I would have thought that ($\dagger$) thus is an element in $\mathit{Ch}(D^+(\mathcal C))$, basically because horizontally, it has the complex property up to homotopy. Does it represent an element in $D^+(\mathcal C)$, or, if you wish, is there a functor $\mathit{Ch}(D^+(\mathcal C)) \to D^+(\mathcal C)$, such that the diagram
$$\begin{matrix}\mathit{Ch}(\mathit{Ch}(\mathcal C)) & \overset{\mathrm{can}_*}{\to} & \mathit{Ch}(D^+(\mathcal C)) \\ \llap{\mathrm{tot}} \downarrow && \downarrow\\ \mathit{Ch}(\mathcal C) & \underset{\mathrm{can}}{\to} & D^+(\mathcal C)\end{matrix}$$ commutes?
Remark. Of course, if I only knew the resolutions $P_{i_{n_k, \bullet}}$ of the $M_{n_k}$'s, I could recover latter from the former and build the reresentative $M_{n_\bullet}$ of $\mathbf LFN$ out of them. But what do I do if I have another functor $F'\colon \mathcal{C} \to \mathit{Ch}(\mathcal C)$ or $F'\colon \mathcal{C} \to D^+(\mathcal C)$ instead, for which the images need not be quasi-isomorphic to complexes concentrated in one degree?