The proof that I am trying to detail is that of Theorem 6.23 of the Book of Abelian Categories by Peter Freyd. The statement is as follows.
*Let $\mathcal{B}$ be a Grothendieck category, $I$ and ordered set, and $\{f_{kj}:E_k\to E_j \}_{k<j}$ a family of monomorphism such that for $k<j<l$, $E_k \to E_j \to E_l =E_k \to E_l$. Then there exits an object $E \in \mathcal{B}$ and a family of monomorphism $\{g_j:E_j \to E\}$ such that for $k<j$, $$E_k \to E_j \to E=E_k \to E$$ *
Sketch of proof: Let $S=\sum \limits_{j\in J} E_j$ be the sum of family $\{E_j\}_{j \in J}$ with canonical inyections $i_j:E_j \to S$. For fix each $j \in J$, by the universal propiety of $S=\sum \limits_{j\in J} E_j$ apliedd to family $E_j=\{i_j \circ f_{kj}:E_k \to S \}_{k<j} \cup \{i_k: E_k \to S \}_{j \leq k} $, exist unique morfism $h_j:S \to S$ such that
for $k<j$, the triangle
commutes
and
for $j\leq k$ the triangle
commutes
Of this construction its follows the nexts statement:
(a)For each $k,j \in J$, el morfismo $h_j \circ i_k:E_k \to E_j$ is a monomorphism.
(b) if $j^\prime<j$, then triangle
is conmutative.
(c) The family $\bigcup \ker(h_j)$ is a ascendent chain with the inclusion of subobjets.
take an epimorphism $h:S\to E$ such that $\ker(h)=\bigcup_k\ker(h_k)$ (it can be for example the cokernel of the subobject $\bigcup_k\ker(h_k)$). For each $j$, let $g_j=h\circ i_j:E_j\to E$. The family formed by the $g_j$ will be the required family. To verify that you must prove that
- the $g_j=h \circ i_j$ are monomorphism. Its follow of items (a),(b) and (c).
- if $k<j$, then the triangle
conmuta.
My question is how show the item 2. I made is the following: To prove that the triangle above commutes, by definition of $g_k$ and $g_j$, I must prove that the diagram
commutes
Now, by definition of $h_j$ and inequality $k<j$, the quadrilateral of the diagram
commutes. Then, to check the conmutative of all the triangle, must show the conmutative of upper triangle. It is $h \circ h_j=h$. My problem is : How show this?