I am trying to compute the fundamental group of a genus-$2$ surface using van Kampen. Let $U_1$ and $U_2$ be the component tori with $U_1 \cap U_2 = U_0$ homotopically equivalent to a circle. I am convinced that $U_1$ is homotopically equivalent to a torus missing a point, which has fundamental group $\mathbb{Z} * \mathbb{Z}$.
So the fundamental group of the genus-2 surface is $\pi_1(U_1) \cdot \pi_1(U_2) / N = \left \langle a, b, c, d \right \rangle / N$, where: $$ N = \{ \left \langle i_{1}(w) i_{2}(w^{-1}) \right \rangle : w \in \pi_1(U_0) \cong \mathbb{Z} \}$$
Here, $i_{1}:\pi_1(U_0) \rightarrow \pi_1(U_1)$ and $i_{2}:\pi_1(U_0) \rightarrow \pi_1(U_2)$ are induced by the inclusions $U_0 \hookrightarrow U_1$ and $U_0 \hookrightarrow U_2$. I have seen the solution so I know what $N$ should be, but I am confused why this is the case. I think my confusion lies in the descriptions of $i_1$ and $i_2$. Does anyone have an intuitive explanation for what is going on here? Any help / advice is appreciated. Thanks