Let
$$
\begin{array}{rccccl}
A\_0&\to& B\_0&\to& C\_0&\to\\
\downarrow & &\downarrow&&\downarrow\\
A\_1&\to& B\_1&\to& C\_1&\to\\
\downarrow & &\downarrow&&\downarrow\\
\vdots & &\vdots&&\vdots\\
\end{array}
$$
be a commutative diagram in a triangulated category such that all the rows are exact triangles. One defines the homotopy colimit of a sequence $A\_0\to A\_1\to\ldots$ as a third object in an exact triangle
$$
\coprod A\_n\xrightarrow{1-shift}\coprod A\_n\to \operatorname{hocolim}\_nA\_\*
$$
which is unique up to (non-unique) isomorphism. For a morphism of diagrams $A\_\*\to B\_\*$ (as above), one gets a (non-unique) morphism $\operatorname{hocolim}\_nA\_\*\to \operatorname{hocolim}\_nB\_\*$.



> 
> Given the diagram above, is there a sequence of morphisms $$
>  \operatorname{hocolim}\_nA\_\*\to \operatorname{hocolim}\_nB\_\*\to
>  \operatorname{hocolim}\_nC\_\*\to $$ which is an exact triangle? I am especially interested in the case that $B\_\*$ is a constant diagram.
> 
> 
> 


I was trying to use the 9-lemma for triangulated categories, but this explicitely **constructs** the morphisms $\coprod C\_n\to \coprod C\_n\to \tilde C$ and I cannot prove, why the first map has to be $1-shift$ and therefore why $\tilde C$ has to be the homotopy colimit of $C\_0\to C\_1\to\ldots$. [This question](https://math.stackexchange.com/questions/692537/question-on-the-fill-in-morphism-in-a-triangulated-category) is strongly related, I think.



Is 'the' homotopy colimit of a sequence of exact triangles an exact triangle?
I'll assume that you at least want your triangulated category to have the property that a (countable) coproduct of exact triangles is an exact triangle?


Even then, I think this is probably not true in a general triangulated category, but it is in many (most?) familiar examples. For example, in the derived category (of an abelian category where coproducts of exact sequences are exact), it's possible to "rigidify" the original directed system of exact triangles to get a directed system of short exact sequences of complexes, so that your question becomes a question about genuine colimits of complexes, rather than homotopy colimits in the derived category.


