I'm interested in the following problem : let $\mathcal{C}$ be an $\infty$-category and $\mathcal{D}:=D_\infty(\mathbb{Z})$ the derived $\infty$-category of abelian groups. Consider functors $A, B, C ,D :\mathcal{C}\to \mathcal{D}$, and natural transformations $B\to C$, $A\to C$ such that $D=Cone(B\to C)$. To construct a natural transformation $A\to B$, it suffices to construct a functorial homotopy from the composite $\tau : A\to C \to D$ to $0$. Are there some known methods to tackle this problem ?
Let me describe some of the approaches I tried. In the specific case I'm studying, we have some information about the homotopy groups $\pi_n \mathrm{Map}(Ax,Dy)$ for $x,y\in \mathcal{C}$, in particular they are trivial except for $n=1,2,3$.
Since we have $\pi_0 \mathrm{Map}(Ax,Dy)=0$ for all $x,y\in\mathcal{C}$, a result of the type "a natural transformation is (homotopic to) $0$ if and only if it is at each point" would suffice, but I don't expect it to hold.
Since the mapping space $\mathrm{Map}(A,D)$ is the end $\int_{c\in\mathcal{C}}\mathrm{Map}(Ac,Dc)$ which is the totalization of a certain cosimplicial object in $\mathcal{D}$, we might be able to use the spectral sequence of a totalization in a stable $\infty$-category from Remark 1.2.4.4 in J. Lurie's "Higher Algebra" ; its first page has rows given by the normalized Moore complex of the cosimplicial abelian groupe formed by the homotopy groups of the cosimplicial object. I don't know how to progress from here since 1) I do not know the exat form of the cosimplicial object and 2) I expect to obtain a fourth quadrant spectral sequence which I don't know how to analyze.
Having learned derived categories from Gelfand & Manin's book, I noticed that in proofs where they construct an homotopy, often the homotopy seems to "write itself". Maybe a brute-force approach, using the specifities of quasicategories, could enable one to construct a functorial homotopy that similarly "writes itself". Are there known examples of a construction of this type ?
Is there maybe a spectral sequence approach for the homotopy groups of the space $\mathrm{Map}_{\mathrm{Map}(A,D)}(\tau, 0)$ ? What I would really like is to have an essentially unique choice, i.e. show that the above space is contractible.
