A free product with amalgamation has the factor presentations plus the amalgamating relations

Given: The objects and hypotheses in the statement.

If $f:K\to G$ and $h:K\to H$ are injective homomorphisms, their pushout is called the free product with amalgamation and is denoted $G\ast_KH$. The quotient construction is thm-group-pushout-as-an-amalgamated-quotient, and injectivity means the trivial-kernel condition of thm-group-homomorphism-injective-iff-trivial-kernel. The notation anticipates identifying $K$ with its two images, but injectivity of the canonical maps $G,H\to G\ast_KH$ is a theorem, not part of this definition. (Free products with amalgamation along monomorphisms).

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).

Suppose each $G_i$ has a presentation $\langle X_i\mid R_i\rangle$, with the alphabets replaced by disjoint copies. Then $\ast_iG_i\cong\left\langle\bigsqcup_iX_i\ \middle|\ \bigcup_iR_i\right\rangle.$ (A free product has the union presentation of presentations of its factors).

Let $F(X)$ be a free group and let $R\subseteq F(X)$ be a set of words, called relations. The group with presentation $\langle X\mid R\rangle:=F(X)/\langle\!\langle R\rangle\!\rangle_{F(X)}$ is the quotient by the normal closure of $R$. The members of $X$ are its generators. In this quotient, every relation in $R$ becomes the identity, as do all consequences forced by normality. (Group presentation by generators and relations).

Let $G$ be a group and $R\subseteq G$. Then $\langle\!\langle R\rangle\!\rangle_G=\left\{g_1r_1^{\varepsilon_1}g_1^{-1}\cdots g_nr_n^{\varepsilon_n}g_n^{-1}:n\in\mathbb N,\ g_i\in G,\ r_i\in R,\ \varepsilon_i\in\{1,-1\}\right\}.$ For $n=0$ the displayed product is the identity. Replacing every conjugator $g_i$ by $g_i^{-1}$ gives the equivalent convention $g_i^{-1}r_i^{\varepsilon_i}g_i$. (The normal closure of $R$ is the set of finite products of conjugates of elements of $R$ and their inverses).