= Solution
In a <pointed category>, a <zero object> defines zero morphisms between all objects. A <categorical cokernel> of $f:A\to B$ is a map $g:B\to C$ with $gf=0$ such that every $u:B\to X$ with $uf=0$ factors uniquely as $u=\bar u g$. Equivalently it is the <coequalizer> of $f$ and the zero map, so $g$ is an <epimorphism>.
Write $a:A\to A'$, $b:B\to B'$, and $c:C\to C'$ for the vertical arrows, with $g'=\operatorname{coker}f'$ and $cg=g'b$. The left <pushout in a category> applied to the compatible pair $g:B\to C$ and $0:A'\to C$ gives $h:B'\to C$ satisfying $hb=g$ and $hf'=0$. The <categorical cokernel> property of $g'$ then gives $d:C'\to C$ with $dg'=h$.
Now $dcg=dg'b=hb=g$. Cancel the <epimorphism> $g$ to obtain $dc=1_C$. To prove the other identity, the two arrows $cdg',g':B'\to C'$ agree after $b$, because $cdg'b=cg=g'b$, and after $f'$, because both composites are zero. The <pushout in a category> uniqueness clause gives $cdg'=g'$. Cancelling the <epimorphism> $g'$ gives $cd=1_{C'}$.
Thus \b[$c$ is an isomorphism]: <cokernel invariance under pushout> holds already in pointed categories with the indicated cokernels.
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