Is there a version of Short Five Lemma in any Exact Category?

Question

It is well known that Short five Lemma (https://en.m.wikipedia.org/wiki/Short_five_lemma) holds in any abelian category.

My question is: Does a version of Short five Lemma also hold in exact category? i.e. Let $(\mathcal A,\mathcal E) $ be an exact category (https://en.m.wikipedia.org/wiki/Exact_category) and let $A\to B \to C$ and $A' \to B' \to C'$ be in $\mathcal E$ such that there is a commutative diagram $$\newcommand\twoheaduparrow{\mathrel{\rotatebox{90}{$\twoheadrightarrow$}}} \begin{array} A & A & {\rightarrow} & B &{\rightarrow} & C & \ \\ & \downarrow{f} & &\downarrow{g}& &\downarrow{h}\\ & A' & \stackrel{}{\rightarrow} &B' & \stackrel{}{\rightarrow} & C' \end{array}$$ where $f,h$ are isomorphisms in $\mathcal A$. Then, is it true that $g$ is an Isomorphism?

If this is known, is there any reference for this result?

Thanks

Answer 1

Every exact category embeds fully and exactly in an abelian category. So just embed this diagram in the abelian category and apply the short five lemma (with the $0$s you’ve left out) there.