= Solution
{c}
Take the <free product> $F=G_1*G_2$, with its natural <group homomorphisms> $k_j:G_j\to F$, and form the <normal closure>
$$
N=\left\langle\!\left\langle
k_1(\phi_1(h))\,k_2(\phi_2(h))^{-1}:h\in G_0
\right\rangle\!\right\rangle.
$$
The <pushout of groups> is the <quotient group>
$$
\boxed{P=(G_1*G_2)/N,}
$$
equipped with the <group homomorphisms> $j_1,j_2$ obtained by composing $k_1,k_2$ with the <quotient map of groups>. The defining <relators> ensure
$$
j_1\phi_1=j_2\phi_2.
$$
Thus the two prescribed images of every <group element> of $G_0$ agree in $P$. This construction works for arbitrary <group homomorphisms> $\phi_1,\phi_2$; no <injectivity> is assumed, and $j_1,j_2$ need not be <injective>. Its <universal property> is proved in the next part.
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