I'm reading "The Geometry of Moduli Space of Sheaves" (Huybrechts,Lehn).On page 26,$Coh_{d}(X)$ is defined as the full subcategory of $Coh(X)$ whose objects are sheaves of dimensions $\le d$.
Let $Coh_{d,d^{'}}(X)$ ($d^{'} \le d$) be the quotient category $Coh_{d}(X)/Coh_{d^{'}-1}(X)$. A morphism$ f: F \rightarrow G$ in $Coh_{d,d^{'}}(X)$ is an equivalence class of diagrams $F \stackrel{s}{\leftarrow} G^{'} \rightarrow G$ of morphisms in $Coh_{d}(X)$ such that $ker(s)$ and $coker(s)$ are at most $(d^{'}-1)$-dimension.
And Then they assert that $G$ and $ F$ are isomorphic in $Coh_{d,d^{'}}(X)$ if they are isomorphic in dimension $d^{'}$.
I think what they mean"they are isomorphic in dimension $d^{'}$" is $F/T_{d^{'}-1}(F) \cong G/T_{d^{'}-1}(G)$.
And "$G$ and $ F$ are isomorphic in $Coh_{d,d^{'}}(X)$ " means that there is $F \stackrel{s}{\leftarrow} G^{'} \stackrel{t}{\rightarrow}G$,$s$ and $t$ are both in the multiplicative system.But how can I derive $F/T_{d^{'}-1}(F) \cong G/T_{d^{'}-1}(G)$ from this? Can you give me some hints? Thanks a lot.