<s>[INST] Understanding the proof that $A_1+A_2\mathop{\longrightarrow}\limits^{\tiny\begin{pmatrix}1&0\\0&1\end{pmatrix}}A_1\times A_2$ is an isomorphism Fix an abelian category. Let $A_1$ and $A_2$ be objects and let $A_1\times A_2$ be their product and $A_1+A_2$ their coproduct (i.e., sum). I am trying to understand the proof of the following theorem in Peter Freyd's "Abelian Categories."

Theorem 2.35 for abelian categories $A_1+A_2\mathop{\longrightarrow}\limits^{\begin{pmatrix}1&0\\0&1\end{pmatrix}}A_1\times A_2$ is an isomorphism.


Proof: Let $K\to A_1+A_2$ be the kernel of $\begin{pmatrix}1&0\\0&1\end{pmatrix}$. Then $K\longrightarrow A_1+A_2\mathop{\longrightarrow}\limits^{\begin{pmatrix}1&0\\0&1\end{pmatrix}}A_1\times A_2\mathop{\longrightarrow}\limits^{p_2}A_2\ =\ K\longrightarrow A_1+A_2\mathop{\longrightarrow}\limits^{\begin{pmatrix}0\\1\end{pmatrix}} A_2$ and $K\longrightarrow A_1+A_2$ is contained in $A_1\mathop{\longrightarrow}\limits^{u_1}A_1+A_2$. Similarly it is contained in $A_2\mathop{\longrightarrow}\limits^{u_2}A_1+A_2$, and hence it is contained in their intersection, which is zero. Thus $K=O$ and $\begin{pmatrix}1&0\\0&1\end{pmatrix}$ is monomorphic. Dually it is epimorphic and hence an isomorphism.

I don't understand the second sentence of the proof. Why are $K\longrightarrow A_1+A_2\longrightarrow A_1\times A_2\longrightarrow A_2$ and $K\longrightarrow A_1+A_2 \longrightarrow A_2$ the same, and why is $K\longrightarrow A_1+A_2$ contained in $A_1\longrightarrow A_1+A_2$?
 [/INST] By definition, $p_2\circ\begin{pmatrix}1&0\\0&1\end{pmatrix} = \begin{pmatrix}0\\1\end{pmatrix}$, so the composites $K\to A_1 + A_2 \to A_1\times A_2 \to A_2 = K\to A_1 + A_2\to A_2$ are the same. But by definition of $K$, $K\to A_1+A_2\to A_1\times A_2$ is zero, so we conclude that $K\to A_1 + A_2\to A_2$ is zero.
By Proposition 2.22 and Theorem 2.31 for abelian categories, $u_1$ is a kernel of $\begin{pmatrix}0\\1\end{pmatrix}$, so the map $K\to A_1\times A_2$ factors though $u_1$, and hence $K\to A_1+A_2$ is contained in $A_1\stackrel{u_1}{\longrightarrow} A_1 + A_2$.
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