Fill in the commutative diagram of exact sequences

Question

I don’t know how to create an exact sequence in the second coulumn...

Can $P_{i}⊕Q_{i}$ be the answer to the red boxes? ($i=1,2,3$)

Answer 1

$\require{AMScd}$ Since $Q_0$ is projective and $g$ is epic, there exists a morphism $t:Q_0\to B$ such that $tg=g\circ t=\varrho_0$. Then we have a commutative diagram with exact rows: \begin{CD} \{0\}@>>>P_0@>[\begin{smallmatrix}1&0\end{smallmatrix}]>>P_0\times Q_0@>\left[\begin{smallmatrix}0\\1\end{smallmatrix}\right]>>Q_0@>>>\{0\}\\ @.@V\delta_0VV@VV\left[\begin{smallmatrix}\delta_0f\\t\end{smallmatrix}\right]V@VV\varrho_0V\\ \{0\}@>>>A@>>f>B@>>g>C@>>>\{0\} \end{CD}