Consider the following commutative diagram in the category of $R$-modules where $R$ is an associative ring with identity and all modules are unital. $ \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex} \newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}}$

$$ \begin{array}{c} 0 & \ra{} & A & \ra{f} & B & \ra{g} & C & \ra{} & 0 \\ & & \da{\gamma} & & \da{\beta} & & \da{\alpha} \\ 0 & \ra{} & D & \ra{k} & E & \ra{h} & F & \ra{} & 0 \end{array} $$

Is there any sufficient and necessary conditions to deduce that there are maps $g':C\to B$ and $h':F\to E$ satisfying the following axioms?

- $gg'=1_C$ and $hh'=1_F$
- $\beta g'=h'\alpha$