I'm following a YouTube video on the usage of Van-Kampen theorem for the torus by Pierre Albin. Around 57:35 he states that the normal subgroup $N$ in $$\pi_1(T^2) \cong \pi_1(A) \ast_{\pi(A \cap B)} \pi_1(B) /N $$ is the image of $\pi_1(C)$ inside $\pi_1(A)$ where $C = A\cap B$. Now Hatcher defines the normal subgroup to be the kernel of the homomorphism $\Phi :\pi_1(A) \ast_{\pi(A \cap B)} \pi_1(B) \to \pi_1(T^2)$ and I'm trying to figure out how these two things coincide.
Also the spaces $A, B$ and $C$ are defined on the video pictorially so apologies for not being able to describe them here. It's stated on the video that $$\pi_1(B)=1$$
From Van-Kampens theorem we get the diagram
so when he says that $N$ is the image of $\pi_1(A \cap B)$ inside $\pi_1(A)$ what is he meaning here? Is there some short exact sequence here that I'm not understanding?
$N$ is not the kernel of $\Phi: \pi_1(A) \ast_{\pi(A \cap B)} \pi_1(B) \to \pi_1(T^2) $ - $\Phi$ is supposed to be an isomorphism and this is the whole point of the Van-Kampen theorem. What $N$ actually is $N$ is the kernel of $\pi_1(A) \ast \pi_1(B) \to \pi_1(T^2)$. Now, since $\pi_1(A) \ast_{\pi(A \cap B)} \pi_1(B) $ is basically $\pi_1(A) \ast \pi_1(B)$ with the images of ${\pi_1(A \cap B)}$ in $\pi_1(A)$ and $\pi_1(B)$ are identified, $\pi_1(B) \simeq 0$ implies that the projection $\pi_1(A) \ast \pi_1(B) \to \pi_1(A) \ast_{\pi(A \cap B)} \pi_1(B) $ just kills the image of ${\pi_1(A \cap B)} \to \pi_1(A)$. Since the codomain is isomorphic to $\pi_1(T)$, this is consistent with saying that this image is the kernel of $\pi_1(A) \ast \pi_1(B) \to \pi_1(T^2)$.