This is from Munkres' Topology page 431:
Theorem 70.2 (Seifert-van Kampen theorem, classical version). Assume the hypotheses of the preceding theorem. Let
$$ j : \pi_1(U, x_0) * \pi_1(V, x_0) \longrightarrow \pi_1(X, x_0) $$
be the homomorphism of the free product that extends the homomorphisms $j_1$ and $j_2$ induced by inclusion. Then $j$ is surjective, and its kernel is the least normal subgroup $N$ of the free product that contains all elements represented by words of the form
$$ (i_{1}(g))^{-1} i_{2}(g)), $$
for $g \in \pi_{1}(U \cap V, x_{0}).$
and by hypotheses of the preceding theorem it means that $X$ is a topological space and $U$, $V$ open subsets of $X$ such that $X = U \cup V$ and $x_0 \in U \cap V$
$j_1:\pi_1(U,x_0)\to\pi_1(X,x_0)$ and $ j_2:\pi_1(V,x_0)\to\pi_1(X,x_0)$ are inclusion induced homomorphisms.
I read the proof and understood it but something is really bothering me and that is how do we know that the homomorphism $j$ mentioned in the theorem even exists?
It tells you: $j_1:U\subset X$ and $j_2:V\subset X$ induce maps on $\pi_1$, $\pi_1(U)\to\pi_1(X)$ and $\pi_1(V)\to\pi_1(X)$, and these are genuine homomorphisms, and if you have two groups $A,B$ and a third group $C$ and homomorphisms $A\to C,B\to C$ then there is a unique associated homomorphism $A\ast B\to C$. In this case with $A=\pi_1(U),B=\pi_1(V),C=\pi_1(X)$, there is a unique homomorphism $j:\pi_1(U)\ast\pi_1(V)\to\pi_1(X)$ associated to $j_1,j_2$.