Given a quiver with $Q_0$ and $Q_1$ the sets of its vertices and edges respectively and a commutative ring $K$, we define a $K$-representation of $Q$ as a pair $$V=\left[ \{V_i\}_{i \in Q_0}, \{V(\alpha)\}_{i \in Q_1} \right]$$ where $V_i$ is a $K$-module for every $i \in Q_0$ and $V(\alpha): V_{s(\alpha)} \longrightarrow V_{t(\alpha)}$ is a homomorphism of $K$-modules.
Now, we can define a category $\operatorname{Rep}_K(Q)$ with the $K$-representations of $Q$ as objects and morphisms $f:V \longrightarrow W$ being a family $f=\{f_i\}_{i \in Q_0}$ such that each $f_i$ is a homomorphism of $K$-modules between $V_i$ and $W_i$ that satisfies $$f_{t(\alpha)} \circ V(\alpha) = W(\alpha) \circ f_{s(\alpha)}$$ for evey $\alpha \in Q_1$.
Now I want to prove the following result:
Given $f:V \longrightarrow W$ a morphism in the category $\operatorname{Rep}_K(Q)$ the following are equivalent:
- $f$ is a section
- $f_i:V_i \longrightarrow W_i$ is a monomorphism in $K-\operatorname{Mod}$ for every $i \in Q_0$ and there exists a $K$-submodule $W_i'$ of $W_i$ such that $$W_i= \operatorname{Im}(f_i) \oplus W_i'$$ and for every $\alpha \in Q_1$ $$W(\alpha)\left(W_{s(\alpha)}'\right) \subset W_{t(\alpha)}'$$
- There exists a $K$-representation $U$ of $Q$ and a morphism $g: U \longrightarrow W$ in $\operatorname{Rep}_K(Q)$ such that the induced morphism in the coproduct $V \sqcup U \longrightarrow W$ is an isomorphism
I've managed to prove that $1$ implies $2$ but I'm having troubles with the other implications.
For $2$ implies $3$ I've defined $$U=\left[ \{W_i'\}_{i \in Q_0}, \{U(\alpha)\}_{i \in Q_1} \right]$$ with $U(\alpha): W_{s(\alpha)}' \longrightarrow W_{t(\alpha)}'$ such that $U(\alpha)(x)=W(\alpha)(x)$. By $2$ is well defined and it is in fact a $K$-representation of $Q$.
Now I've also defined $g:U \longrightarrow W$ such that $g_i$ is the inclusion of $U_i=W_i'$ in $W_i$ for every $i \in Q_0$ which it is indeed a morphism in $\operatorname{Rep}_K(Q)$.
I think that this objects are the one that will satisfy $3$ but I'm not able to prove it.
What I want to prove is that if $h: V \sqcup U \longrightarrow W$ is the unique morfism in $\operatorname{Rep}_K(Q)$ such that $$f=h \circ q_1$$ $$g=h \circ q_2$$ where $q_1: V \longrightarrow V \sqcup U$ and $q_2: U \longrightarrow V \sqcup U$ are the injections of the coproduct then this is an ismorphism.
To do that I've defined $\bar h: W \longrightarrow V \sqcup U$ such that $\bar h_i: W_i \longrightarrow (V \sqcup U)_i$ is given by $$\bar h_i (x)= \left\{ \begin{array}{lcc} (q_1)_i\left( f_i^{-1}(x) \right) & \mathrm{if} & x \in \operatorname{Im}(f_i) \\ (q_2)_i\left( g_i^{-1}(x) \right) & \mathrm{if} & x \in W_i' \end{array} \right.$$
and I was trying to prove it is the inverse of $h$, but I don't know how to do it.
Any ideas of how to prove this last part? Or, if it is not the correct way, any idea of how to prove this implication?
For the implication of $3$ to $1$ I don't even know how to start so any help there would also be really apreciated.