I'm studying homological algebra. And I'm working on this problem.
The relation between thick subcategories in $\mathcal{D}$ and localizing classes of morphisms is as follows. A localizing class $S$ in $\operatorname{Mor}(\mathcal{D})$ is said to be saturated if \begin{align*} &\, s \in S \\ \iff&\, \text{there exist morphisms $f,f'$ in $\mathcal{D}$ such that $f \circ s \in S$ and $s \circ f' \in S$}. \end{align*}
Prove the following result. Let $\mathcal{D}$ be a triangulated category. Then $$ \mathcal{C} \mapsto \varphi(\mathcal{C}) = \left\{ \begin{matrix} \text{$s \in \operatorname{Mor} \mathcal{C}$, $s$ is contained in a distinguished} \\ \text{triangle $X \xrightarrow{s} Y \to Z \to X[1]$ with $Z \in \operatorname{Ob} \mathcal{C}$.} \end{matrix} \right\} $$ determines a one-to-one correspondence between the set of thick subcategories in $\mathcal{D}$ and the set of saturated localizing classes in $\mathcal{D}$ compatible with triangulation.
The converse mapping associates with a class $S \subset \operatorname{Mor} \mathcal{D}$ the full subcategory $\psi(S)$ generated by such objects $Z \in \operatorname{Ob} \mathcal{D}$ that there exists a distinguished triangle $X \xrightarrow{s} Y \to Z \to X[1]$ with $s \in S$.
Let $\mathcal{D}$ be a triangulated category. Then there's a bijection between the set of thick subcategories in $\mathcal{D}$ and the set of saturated localizing classes (sometimes called saturated multiplicative system). This is also known for the Verdier correspondence.
I've already proved that a thick triangulated subcategory can map to a saturated localizing classes by $\varphi$. But I'm stuck with the converse mapping. That is, if $S$ is a saturated localizing classes, then verify $\psi(S)$ being a thick triangulated subcategory.
My Effort: Let $\mathcal{C} = \psi(S)$.
TS1: First verify that $\mathcal{C}$ is closed under $\Sigma$, i.e. $\Sigma \mathcal{C} = \mathcal{C}$. Let $Z \in \operatorname{Ob}(\mathcal{C})$, then there's a distinguished triangle $\require{AMScd}$ \begin{CD} X @>s>> Y @>>> Z @>>> \Sigma X \end{CD} with $s \in S$. So it suffices to show that $\Sigma^n s \in S$. But I don't know how to do this.
TS2: Let $\require{AMScd}$ \begin{CD} X @>>> Y @>>> Z @>>> \Sigma X \end{CD} be a distinguished triangle. Verify that if $X, Y \in \operatorname{Ob}(\mathcal{C})$, then so is $Z$. Actually I'm more confused with this part for I don't know how to use the properties of localizing and saturated.
Can anyone enlighten me? Thanks in advance.