# morphism of monoids and group

let $B,C$ be commutative monoids, $A$ be an abelian group and $B \stackrel{b}{\leftarrow} C \stackrel{a}{\rightarrow} A$ morphisms of monoids. Let $P$ be the push-out of $A$ and $B$ w.r.t. $C$ in the category of monoids. Here my question: Do I get a group homomorphism $$P^{gp}\rightarrow A\oplus B^{gp}$$

Thank you.

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I would rather expect the quotient modulo the image of $C$ in $A\oplus B$ (groupified) on th eright hand side. –  Hagen von Eitzen Jan 8 '13 at 16:11