# Exercise about exact sequence and pushout

abelian-categoriescategory-theoryhomological-algebra

The following is a commutative diagram in an abelian category. Assume that the rows are exact and that $$h,k$$ are epic.
$$\require{AMScd}$$
$$\begin{CD} 0@>>>a@>{f}>> b @>{g}>> c@>>> 0\\ @VVV @VVjV @VVhV @VVkV @VVV\\ 0@>>>a'@>{f'}>> b' @>{g'}>> c'@>>> 0 \end{CD}$$

How do I prove that $$j$$ is epic if the square $$g,h,k,g'$$ is cocartesian? I proved the converse already, but I'm stuck in this direction. Thanks in advance.

EDIT. I post this solution for completeness, since it seems different from the one given in the answer. I didn't use that $$h,k$$ are epic, and this hypothesis isn't useful in proving the converse of sentence in italics either, so I don't know why it was in text of the exercise.

Let $$z:a'\to d$$ be an arrow such that $$zj=0$$, and construct the pushout: $$\begin{CD} a'@>{f'}>> b'\\ @VVzV @VVp_1V \\ d@>{p_2}>> p \end{CD}$$ since $$p_1hf=0$$ and $$g=\operatorname{coker}f$$, there is $$u:c\to p$$ s.t. $$ug=p_1h$$. Being $$g,h,k,g'$$ a pushout, there is $$v:c'\to p$$ s.t. $$vg'=p_1$$, so that $$p_2z=p_1f'=vg'f'=0$$. Since $$p_2$$ is monic (it is the pushout of $$f'$$, that is monic), $$z=0$$.

Consider the following commutative diagram: $$\begin{CD} a@>>>0\\ @VfVV @VVV\\ b @>{g}>> c\\ @VVhV @VVkV\\ b' @>{g'}>> c' \end{CD}$$ By exactness of top row, $$g$$ is the cokernel of $$f$$, hence the top square is a pushout. By pushout pasting lemma, the bottom square is a pushout if and only if the outer rectangle is a pushout as well. The outer rectangle is a pushout if and only if $$g'$$ is the cokernel of $$fh$$. By exactness of the top row, $$f'$$ is the kernel of $$g'$$. Consequently, $$g'$$ is the cokernel of $$fh$$ if and only if $$f'$$ is the image of $$fh$$. Finally, $$j$$ is epic if and only if $$f'$$ is the image of $$jf'$$, hence that of $$fh$$.