Question
The differentials on the $(r+1)$th page of a spectral sequence are maps on the cohomologies of the complexes on $r$th page. So, between two adjacent complexes $K^\bullet,L^\bullet$ on the $r$th page, we have (maybe shifted, depending on indexing conventions) maps $H^n(K^\bullet)\to H^n(L^\bullet)$. Are these maps induced by a map $K^\bullet\to L^\bullet$ in the derived category?
"Evidence"
I think this might be true because, roughly, maps in this spectral sequence can be constructed via snake lemma/connecting map type arguments, and at least for a SES of complexes $0\to A^\bullet \to B^\bullet \to C^\bullet \to 0$, the connecting map $\delta^n: H^n(C^\bullet) \to H^n(A^\bullet[1])$ lifts to a map in the derived category $\delta: C^\bullet \to A^\bullet[1]$.
There's an affirmative answer to the question at least for $r=0$. If we call $(C^{\bullet,\bullet},d_\rightarrow,d_\uparrow)$ our double complex and $({}_{\uparrow}E_r^{\bullet,\bullet},d_r)$ our spectral sequence (where $d_r: E_r^{p,q}\to E_r^{p+r,q-r+1}$), then the maps $$d_1: E_1^{p,q} \to E_1^{p+1,q}$$ are exactly $$H^p(d_\rightarrow): H^p(C^{p,\bullet}).$$
