First gives some definitions, and then the property that I am confused.
$A$, $B$ are both $R$-module, and $C$, $D$ an (additive) abelian group, consider the category $M(A,B)$ whose objects are all middle linear maps on $A\times B$.
Then by definition a morphism in $M(A,B)$ from $f:A\times B \rightarrow C$ to $g:A\times B \rightarrow D$ is a group homomorphism $h:C\rightarrow D$ with $g=h(f)$.
Prop: $h$ is an equivalence in $M(A,B)$ if and only if $h$ is an isomorphism of groups.
I think $h$ only needs to be an "isomorphism" on "image of $f$" to "image of $g$" in order to be an equivalence. Since outside of it there is no restriction given by $f$ and $g$.
For example let $C=A \times B\times X$ and $D=A \times B\times X$ and let $f$ and $g$ be " forgetful inclusion" then $h$ can be arbitrary assign on $X$, hence may not be an isomorphism.
What am I missing?
Reference:
The Definitions of middle linear and the definition of equivalence.
I somehow find the solution myself.
The example I gave is just wrong, it is not middle linear. (Thanks to @Jeremy Rickard for noticing.)
And since the identity of morphism is unique (Thanks to @roman for a remind), $1_{f}$ must be the identity isomorphism on $C$. And similarly for $1_{g}$.
Hence for $h$ to be an equivalence, there must exist $h'$, such that $h'(h)=1_{f}=1_{C}$ and $h(h')=1_{g}=1_{D}$, hence $h$ is an isomorphism from $C$ to $D$.