1) Prove that a diagram like below (of modules) (tickzcd doesn't work? here's (big) pdf, its an exercise in section 1.4 ) with $\pi_{1}'i_{1}' = 1$, $\pi_{2}'i_{2}' = 1$, $\pi_{1}'i_{2}' = 0$, and $(i_{1}', \pi_{2}')$ exact is a direct sum diagram.
\begin{tikzcd} A \arrow[r, "i_{1}'", yshift = .4ex] & D \arrow[l, "\pi_{1}'", yshift = -.4ex] \arrow[r, "\pi_{2}'" swap, yshift = -.4ex] & B \arrow[l, "i_{2}'" swap, yshift = .4ex] \end{tikzcd}
I'd like to prove it elegantly, but I'm not sure how to prove the identity identity from exactness and what's given, short of checking it doesn't mess with all the $\pi's$ and $i's$, which I assume would be enough, but I think there's a nicer way.