In an abelian category, is it true that $$0\to A\stackrel f\to B\stackrel g\to C\to 0$$ being exact means that $C=\operatorname{coker}(f)$?
I checked it in the module category but am having trouble in an arbitrary abelian category.
First isomorphism theorem $B/A=B/\operatorname{im}(f)\cong B/\ker(g)=C$
where $B/A:=\operatorname{coker}(f)$ for $f:A\hookrightarrow B$ is the definition of a quotient object I think