It's well known that chain complexes are an abelian category, and in particular we can consider chain complexes of chain complexes, i.e. *double complexes*. Given a double complex $A^{\bullet\bullet} \in \mathrm{Kom}(\mathrm{Kom}(\mathcal A))$ we can form the total complex $\newcommand{\tot}{\mathrm{Tot}}\tot(A^{\bullet\bullet})$ which now lies "one level lower", in $\mathrm{Kom}(\mathcal A)$.

I can also try to consider chain complexes in the derived category of $\mathcal A$, but it is no longer clear (at least to me) how to build a total complex in the best way, since $d \circ d=0$ only has to hold up to homotopy in $D(\mathcal A)$.

But let me now consider more generally a triangulated category $\newcommand{\T}{\mathcal T}\T$ and a sequence of objects $A^0, \ldots, A^n$ with $d_i \colon A^i \to A^{i+1}$ such that $d_{i+1} \circ d_i = 0$. It seems to me that one can define a total complex $\tot(A^\bullet) \in \T$ as an iterated mapping cone: for instance, if $n=2$, then one can first consider $B = \mathrm{Cone}(d_0)$. Then we consider the diagram $$ \begin{matrix} A^0 & \to & A^1 & \to & B \\ \downarrow & & \downarrow & & \\ 0 & \to & A^2 & \to & A^2 & \end{matrix}$$ where both rows are distinguished triangles; by one of the axioms of triangulated categories there is a map $f \colon B \to A^2$ completing this to a map of triangles, and we define $\tot(A^\bullet)=\mathrm{Cone}(f)$.

However, this has a the drawback of not being functorial. For instance, I think that one would like to say that a map $A^\bullet \to B^\bullet$ of chain complexes in $\T$ is a quasi-isomorphism if $\tot(A^\bullet) \to \tot(B^\bullet)$ is an isomorphism, but this makes no sense unless $\tot$ is a functor. And one would like to say that if $f \colon \T \to \T'$ is a triangulated functor, then there is a natural equivalence between $f \circ \tot$ and $\tot \circ f$ as functors $\mathrm{Kom}(\T) \to \T'$ (am I right?), and again $\tot$ needs to be functorial.

First of all, I would like to know if what I've said so far is correct. Maybe there is a better way to set things up than this? Secondly, I've heard the slogan that stable $\infty$-categories solve all problems arising from the fact that triangulated categories don't have functorial mapping cones. Is there a better behaved notion of a chain complex in a stable $\infty$-category?

I suspect that I could answer these questions myself if I started reading Lurie's work, but it's a slightly intimidating amount of text and I thought I'd ask here first.