I am currently trying to understand why finite products and coproducts in the category $\text{Ab}$ coincide. In fact, I'm not even sure I can show it. My question is the following:
Is there an intuitive way to understand why finite products and coproducts in $\text{Ab}$ coincide, while the same is not true in $\text{Grp}?$
As for the formal direction, I'm not sure if I have shown that the two coincide. The idea is to show that $G\times H$ satisfies the universal property for coproducts for $G,H \in \text{Obj}(\text{Ab})$.
Attempt
I defined the inclusion functions $\iota_G:G\longrightarrow G\times H$ and $\iota_H:H\longrightarrow G\times H$ by $\iota_G(g)=(g,1_G)$ and $\iota_H(h)=(1_H,h)$. Suppose $\varphi:G\longrightarrow K$ and $\psi:H\longrightarrow K$ are homomorphisms. To show $G\times H$ is a coproduct in $\text{Ab}$, I then need to construct a unique function $\tau:G\times H\longrightarrow K$ such that $\varphi=\tau\circ\iota_G$ and $\psi=\tau\circ\iota_H$. As such, I defined by $\tau$ by $\tau(g,1_H)=\varphi(g)$ and $\tau(1_G,h)=\psi(h)$. I think this should work since for all $(g,h) \in G\times H$ we have $(g,h)=(g,1_H)*(h,1_H)$, so by defining $\tau$ so that $\tau(g,h)=\tau(g,1_H)*\tau(h,1_H)$ everything should work.