$\def\D{\mathcal{D}}
\def\ind{\operatorname{Ind}}
\def\K{\mathcal{K}}
\def\A{\mathcal{A}}$Inside [GW, Remark F.168, p. 794], we find:

> [Let $\K$ be a category  and let $\K_S$ be its localization with respect to a right multiplicative system $S$.] If $\iota_\K:\K\to\ind(\K)$ is the fully faithful canonical functor and $Q:\K\to\K_S$ is the localization functor, one obtains a diagram of functors
$$
\begin{matrix}
\K&\xrightarrow{Q}&\K_S\\
\iota_\K&\searrow&\downarrow&\alpha\\
&&\ind(\K)
\end{matrix}
$$
which is *not* commutative. But there exists a natural morphism
$$
\tag{F.42.1}\label{nat}
\iota_\K\to\alpha\circ Q.
$$
Let us suppose that $\K$ is a triangulated subcategory of $K(\A)$ for an abelian category $\A$ and that $\K_S$ is the localization by the system $S$ of quasi-isomorphisms in $\K$. Then $\ind(\K)$ is also triangulated, all functors above are triangulated and the morphism \eqref{nat} is a morphism of triangulated functors.

Here $\alpha:\K_S\to\ind(\K)$ is the canonical embedding of the localization of a category into its ind-completion, see e.g. [KS, Proposition 7.4.1]. Namely, $\alpha$ is the left Kan extension of $\iota_\K$ along $Q$ [KS, Proposition 7.3.3].

My questions are:

**(Q1)**. Given a triangulated category $\K$, is there some canonical triangulated structure on its ind-completion $\ind(\K)$ turning $\K\to\ind(\K)$ into a triangulated functor?

**(Q2)**. If the answer is to (Q1) is 'no' in general but 'yes' if $\K$ is a full subcategory of $K(\A)$, where $\A$ is abelian (as [GW] claims in the quote above), do you know any reference for this? And same for [GW]'s claims that $\iota_K$ and $\alpha$ are triangulated functors and \eqref{nat} is a morphism of triangulated functors. In [KS] there's nothing on these issues.

----------

Context: I was trying to find a proof of the fact that derived triangulated functors are triangulated in terms of ind-completions. Despite the negative answer to (Q1) that Neil Strickland provided, I found a workaround for the proof that I explain [here][1].

### References ###

[GW]. Görtz, Wedhorn, *Algebraic Geometry II*

[KS]. Kashiwara, Schapira, *Categories and Sheaves*


  [1]: https://mathoverflow.net/a/479865/101848