I would like to understand the accounts of P. Gabriel (link text), pag 365, when he shows that the composition of this category is well defined.

Definition: Given a Serre subcategory $\mathcal{C}$ of $\mathcal{A}$, the quotient category $\mathcal{A}/\mathcal{C}$ of $\mathcal{A}$ with respect to $\mathcal{C}$ is defined as follows. The objects in $\mathcal{A}/\mathcal{C}$ are the objects in $\mathcal{A}$. Given two objects $A,B$ in $\mathcal{A}$, there is for each pair of subobjects $A' \subset A$ and $B'\subset B$ and induced map $Hom_{\mathcal{A}}(A,B) \to Hom_{\mathcal{A}}(A',B/B')$. The pairs $(A',B')$ such that both $A/A'$ and $B'$ lie in $\mathcal{C}$ form a directed set, and one obtains a direct system of abelian groups $Hom_{\mathcal{A}}(A',B/B')$. We define $$ Hom_{\mathcal{A}/\mathcal{C}} (A,B)= colim_{(A',B')} Hom_{\mathcal{A}}(A',B/B') $$ Let $\bar{f} \in Hom_{\mathcal{A}/\mathcal{C}} (M,N)$ and $\bar{g} \in Hom_{\mathcal{A}/\mathcal{C}} (N,P)$. Else, exist $f \in Hom_{\mathcal{A}}(M',N/N')$ and $g \in Hom_{\mathcal{A}}(N'',P/P')$. Thus, let $M''=f^{-1}((N''+N')/N')$. Why $M/M'' \in \mathcal{C}$? (first question).

The author then takes the following object: $g(N'' \cap N')$ lie in $\mathcal{C}$. Why? (second quastion).

Let $P''=P'+ g(N'' \cap N')$, too $P'' \in \mathcal{C}$, why? (third question)

Now, let the morphism $f' : M'' \to (N''+N')/N'$ induced $f$ and the morphism $g': N''/(N''\cap N') \to P/P''$ induced $g$.

Finally, it takes the following morphism: $M'' \xrightarrow{f'} (N''+N')/N' \cong N''/(N''\cap N') \xrightarrow{g'} P/P''$. He claims that this morphism is the representative of composite function $\bar{g}\bar{f}$ independently of morphisms $f$ and $g$. I do not understand this statement. (fourth question).

Comments: Let $f: A \to B$ and $A' \subset A$, $f(A')$ is the image of the $ A'\hookrightarrow A \xrightarrow{f} B$. If $B'\subset B$, $f^{-1}(B')$ is the kernel of the $ A \xrightarrow{f} B \twoheadrightarrow B/B'$.