Is the ind-completion of a triangulated category triangulated?

Question

$\def\D{\mathcal{D}} \def\ind{\operatorname{Ind}} \def\K{\mathcal{K}} \def\A{\mathcal{A}}$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].

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.

References

[GW]. Görtz, Wedhorn, Algebraic Geometry II

[KS]. Kashiwara, Schapira, Categories and Sheaves

Answer 1

The category $\mathcal{F}$ of finite spectra is self-dual under Spanier-Whitehead duality, so $\operatorname{Pro}(\mathcal{F})\simeq\operatorname{Ind}(\mathcal{F})$. The $\operatorname{Pro}$ category is equivalent to the category of homology theories, or equivalently spectra mod phantoms, by Proposition 4.19 of Morava K-theories and localisation. Of course the category of spectra is triangulated, but I am fairly sure that the quotient spectra/phantoms is not, although I do not immediately see a proof. Although the cited proof is written for the category of finite spectra, it will apply much more generally, although some kind of countability hypothesis may be required (as in Theorem 4.1.5 of Axiomatic stable homotopy).

On the other hand, if you interpret everything in Lurie's framework, I think it is true that the $\operatorname{Ind}$ completion of any stable $\infty$-category is a stable $\infty$-category, and that is in some sense the right version of your question.