The kernels of the amalgamating maps are killed in the opposite canonical maps to a group pushout

Given: The objects and hypotheses in the statement.

Given homomorphisms $f:K\to G$ and $h:K\to H$ as in def-group-homomorphism, a pushout is a group $P$ with homomorphisms $i_G:G\to P$ and $i_H:H\to P$ such that $i_G\circ f=i_H\circ h$, and such that every compatible pair $u:G\to Q$, $v:H\to Q$ factors through a unique $w:P\to Q$ with $w\circ i_G=u$ and $w\circ i_H=v$. The maps $f,h$ need not be injective. (Pushouts of group homomorphisms).

For homomorphisms $f:K\to G$ and $h:K\to H$, let $N$ be the normal closure in $G\ast H$ of $\{j_G(f(k))j_H(h(k))^{-1}:k\in K\}.$ Then $(G\ast H)/N$, with the induced factor maps $j_G$ and $j_H$, is a pushout of $f$ and $h$. (A group pushout is the quotient of a free product by the amalgamating relations).

Let $G$ and $G'$ be groups with identities $e$ and $e'$, and let $f : G \to G'$ be a group homomorphism (def-group-homomorphism), so $f(xy) = f(x)f(y)$ for all $x, y \in G$. Then: 1. $f(e) = e'$; 2. $f(g^{-1}) = f(g)^{-1}$ for every $g \in G$; 3. $f(g^{n}) = f(g)^{n}$ for every $g \in G$ and every $n \in \mathbb{Z}$, powers being those of def-group-power. For monoid homomorphisms the analogue of claim 1 is false, so preservation of the identity has to be part of the definition: the map $u : \mathbb{Z} \to \mathbb{Z}$ with $u(x) = 0$ for every $x$ satisfies $u(xy) = u(x)u(y)$ for the multiplicative monoid $(\mathbb{Z},\cdot,1)$, yet $u(1) = 0 \ne 1$. (A group homomorphism automatically satisfies $f(e) = e'$ and $f(g^{-1}) = f(g)^{-1}$, and $f(g^{n}) = f(g)^{n}$ for every $n \in \mathbb{Z}$; for monoid homomorphisms preservation of the identity must be assumed).