$\newcommand{Ab}{\operatorname{Ab}} \newcommand{Id}{\operatorname{Id}}$I'm self-studying Introduction to Topological Manifolds by John M. Lee, which includes quite a few exercises like this:
9-4(b) Let $S_1$ and $S_2$ be disjoint sets, and let $R_i$ be a subset of the free group $F(S_i)$ for $i=1,2$. Prove that $\langle S_1 \cup S_2 \mid R_1 \cup R_2 \rangle$ is a presentation of the free product group $\langle S_1 \mid R_1 \rangle * \langle S_2 \mid R_2 \rangle$.
10-17. For any groups $G_1$ and $G_2$, show that $\Ab(G_1*G_2) \cong \Ab(G_1) \oplus \Ab(G_2)$.
10-19. For any set $S$, show that the abelianization of the free group $F(S)$ is isomorphic to the free abelian group $\mathbb{Z}S$.
($*$ is free product, $\Ab$ is abelianization.) Here is my tedious proof of 10-17:
For $i=1,2$, let \begin{align*} \alpha_{i}:G_{i} & \to\Ab(G_{i}),\\ \alpha:G_{1}*G_{2} & \to\Ab(G_{1}*G_{2}),\\ j_{i}:\Ab(G_{i}) & \to\Ab(G_{1})\oplus\Ab(G_{2}),\\ k_{i}:G_{i} & \to G_{1}*G_{2} \end{align*} be the canonical maps. There exists a homomorphism $\ell:G_{1}*G_{2}\to\Ab(G_{1})\oplus\Ab(G_{2})$ satisfying $\ell\circ k_{i}=j_{i}\circ\alpha_{i}$, and there exists a homomorphism $\varphi:\Ab(G_{1}*G_{2})\to\Ab(G_{1})\oplus\Ab(G_{2})$ satisfying $\varphi\circ\alpha=\ell$. Also, there exist homomorphisms $m_{i}:\Ab(G_{i})\to\Ab(G_{1}*G_{2})$ satisfying $m_{i}\circ\alpha_{i}=\alpha\circ k_{i}$, so there exists a homomorphism $\psi:\Ab(G_{1})\oplus\Ab(G_{2})\to\Ab(G_{1}*G_{2})$ satisfying $\psi\circ j_{i}=m_{i}$. Now $ \varphi\circ\psi\circ j_{i}\circ\alpha_{i}=\varphi\circ m_{i}\circ\alpha_{i}=\varphi\circ\alpha\circ k_{i}=\ell\circ k_{i}=j_{i}\circ\alpha_{i}, $ so $\varphi\circ\psi=\Id_{\Ab(G_{1})\oplus\Ab(G_{2})}$ by uniqueness. Similarly, $ \psi\circ\varphi\circ\alpha\circ k_{i}=\psi\circ\ell\circ k_{i}=\psi\circ j_{i}\circ\alpha_{i}=m_{i}\circ\alpha_{i}=\alpha\circ k_{i}, $ so $\psi\circ\varphi=\Id_{\Ab(G_{1}*G_{2})}$ by uniqueness.
This is just one example - there are many other proofs which seem to follow the same pattern - use the universal properties to derive homomorphisms, compose a bunch of them together, simplify, and prove that you have isomorphisms. My questions are:
- Is there a name for this kind of proof?
- How can I understand what I'm doing? I feel like I'm just moving symbols around right now, matching up domains/codomains.
- Can these proofs be made less tedious?
- The reason I say these proofs are like "magic" is that everything seems to fit together perfectly when I'm composing the maps. Why does this happen?