ScalingStacks

Proof: Properties (i) and (ii) are immediate consequences of the definition of a co-$t$-structure: the ${\mathrm{\Sigma }}^{-1}𝒜$-precover $\alpha :{\mathrm{\Sigma }}^{-1}A\to X$ and the $ℬ$-preenvelope $\beta :X\to B$ are just the first and second morphisms in the distinguished triangle given by property $(3)$ of Definition 1.4. Property (iii) is a consequence of condition $(0)$ and the orthogonality condition $(2)$ of Definition 1.4, and properties (iv) and (v) are easy consequences of property (iii), see [12]. $\mathrm{\square }$