Almost split morphisms in subcategories of triangulated categories

For a suitable triangulated category $\mathcal{T}$ with Serre functor $S$ and full precovering subcategory $\mathcal{C}$ closed under summands extensions, an indecomposable object $C$ in is called Ext-projective if Ext$^1(C,\mathcal{C})=0$. Then there no Auslander-Reiten triangle end term $C$. In this paper, we show that if, for such $C$, minimal right almost split morphism $\beta:B\rightarrow C$ $\mathcal{C}$, then appears something very similar to $\mathcal{C}$: essentially unique of the form \begin{align*} \Delta= X\xrightarrow{\xi} B\xrightarrow{\beta} C\rightarrow \Sigma X, \end{align*} where $X$ not $\xi$ $\mathcal{C}$-envelope $X$. Moreover, some extra assumptions, removing from replacing it produces new extensions. We prove process coincides classic mutation respect rigid generated by all Ext-projectives apart When cluster Dynkin type $A_n$ has above properties, give description triangles $\Delta$ which circumstances gives extension subcategory.