Let $\mathscr{A},\mathscr{B}$ be abelian categories, the first with enough projectives, together with a right-exact functor $F\colon \mathscr{A}\to\mathscr{B}$ (in my example, it is a tensor product, but I am interested in the general setting). Let $\mathcal{D}^-(\mathscr{A}),\mathcal{D}^-(\mathscr{B})$ be their (bounded-above) derived categories and $\mathbb{L}F$ the left-derived functor of $F$. Given a chain complex $X_\bullet\in\textbf{comp}(\mathscr{A})$ we can construct an edge-map from a certain spectral sequence and another morphism coming from functoriality of the derived-category machinery: I need a reference showing that these two maps actually coincide. Let me go into the details of their construction.
Given a chain complex $X_\bullet\in\textbf{comp}(\mathscr{A})$ together with a Cartan–Eilenberg projective resolution $Q_{\bullet\bullet}\to X_\bullet$, we can define the total complex $P_\bullet=\mathrm{Tot}(Q_{\bullet,\bullet})$ (assume all needed boundness, existence of sums, etc.): it gives rise to a quasi-isomorphism $\epsilon\colon P_\bullet\to X_\bullet$ on which I can then perform two operations, and I'd like to show that they coincide.
The first is to consider the composition of (the extension to complexes of) $F$ with $q_\mathscr{B}\colon \textbf{comp}(\mathscr{B})\to\mathcal{D}^-(\mathscr{B})$: applying this to $\epsilon\colon P_\bullet \to X_\bullet$ I obtain $(q_\mathscr{B}\circ F )(\epsilon)\colon F(P_\bullet)\longrightarrow F(X_\bullet)$ (as a morphism in the derived category $\mathcal{D}^-(\mathscr{B})$). By inverting signs, in order to turn chain complexes into cochain ones, and applying the cohomological functor $H^n\colon\mathcal{D}^+(\mathscr{B})\to\mathbf{Ab}$, I end up with a map $\alpha_1$: $$ (H^n\circ q_\mathscr{B}\circ F)(\epsilon)\colon H^n(F(P^\bullet))=(\mathbb{L}^nF)(X^\bullet)\overset{\alpha_1}{\longrightarrow}H^n(F(X^\bullet)). $$
The second is the edge-map in the corresponding spectral sequence, built as follows. By applying $F$ to the Cartan–Eilenberg resolution we obtain a double complex $F(Q_{\bullet\bullet})$: by construction, this is a second quadrant double complex, but since I prefer to do cohomology rather than homology, I invert signs and consider $F(Q_{-\bullet,-\bullet})$: it is a fourth quadrant double complex. This we can filter by its "second degree" to obtain a fourth quadrant spectral sequence $E_2^{p,q}=H^p(F(Q^{\bullet q}))$ which converges to the hypercohomology of $X^\bullet$, namely $$ E_2^{p,q}=H^p(F(Q^{\bullet,q}))\Longrightarrow \mathbb{L}^{p+q}(X^\bullet). $$ Corresponding to this spectral sequence there is a edge morphism obtained as follows. Since the spectral sequence is concentrated in the fourth quadrant, all differentials landing in the term $E_{2}^{n,0}$ are $0$: it follows that $E_r^{n ,0}$ is a subobject of $E_2^{n,0}$ for all $r\geq 2$ (it is the kernel of the differential $d_{r-1}^{n,0}$) and since the sequence stabilizes we obtain a map $\beta\colon E_\infty^{n,0}\to E_2^{n,0}$. On the other hand, convergency means that $\mathbb{L}^{p+q}(X^\bullet)$ has a dicreasing filtration $\mathrm{Fil}^j(\mathbb{L}^{n}F)(X^\bullet)$ with associated grading $E_\infty^{j ,n-j}$: in particular, $\mathrm{Fil}^{n}(\mathbb{L}^{n}F)(X^\bullet)=\mathbb{L}^{n}(X^\bullet)$ and $E_\infty^{n,0}=(\mathbb{L}^{n}F)(X^\bullet)/\mathrm{Fil}^{n+1}\mathbb{L}^{n}(X^\bullet)=(\mathbb{L}^{n}F)(X^\bullet)$ is a quotient of $(\mathbb{L}^{n}F)(X^\bullet)$. Composing this quotient map with the map $\beta$ constructed above gives us the edge map $\alpha_2$: \begin{equation} \alpha_2\colon (\mathbb{L}^{n}F)(X^\bullet)\longrightarrow E_\infty^{n,0}\longrightarrow E_2^{n,0}=H^n(F(Q^{\bullet,0})=H^n(F(X^\bullet)) \end{equation}
I am looking for a proof, or a reference, that $\alpha_1=\alpha_2$.
