Let $\mathcal{A}$ be an additive category with countable coproducts. I am just starting to learn about homotopy colimits and I am struggling with the following example that I am very interested in understanding (you can find this as example 2 of section 4.4 in this paper):
Given an $\mathcal{A}$-complex $X$, we let $X'$ denote the homotopy colimit of $$\tau_n X\to \tau_{n-1} X\to \tau_{n-2} X\to\ldots$$ where $$(\tau_n X)^p:=\left\{\begin{array}{cl} X^p & p\geq n\\ 0 & p<n\end{array}\right. ,$$ $\partial_{\tau_nM}^p=0$ for $p<n$ and $\partial_{\tau_nM}^p=\partial_{M}^p$ for $p\geq n$, and the maps are just the inclusions of the truncations (I could be wrong about this, but I think these are the maps based on the paper).
The part I am having a hard time justifying is that $X'$ is isomorphic to $X$ in $K(\mathcal{A})$, the homotopy category of $\mathcal{A}$. Is there an easy way to see this, I honestly have no idea where to start when showing this? Thanks for any help!