Given a commutative diagram in a Grothendieck category $\mathscr{A}$
\begin{array}{ccccccccc} 0 & \longrightarrow & A' & \overset{i}{\longrightarrow} & A & \overset{p}{\longrightarrow} & A'' & \longrightarrow & 0\\ & & f\downarrow & & g\downarrow & \\ 0 & \longrightarrow & B' & \overset{j}{\longrightarrow} & B & \overset{q}{\longrightarrow} & B'' & \longrightarrow & 0 \end{array}
with $f: A' \rightarrow B'$ and $g: A \rightarrow B$ , There exists a unique arrow $h: A'' \rightarrow B''$ making the large diagram commute.
My question is, if f and g are monic, does this force h to also be monic?
If so .. why? and if not would assuming g to be an isomorphism do the trick?