Explanation of proof in Representation Theory: A Homological Point of View in the book Representation Theory: A Homological Point of View Proposition 3.1.18 Zimmerman proves that a cokernel is a colimit, but I can't understand his proof.
He lets $\left((M_i)_{i \in \mathbb{N}}), \epsilon_{ij}\right)$ be a codirected system and then defines a map
\begin{align}
\varphi : &\coprod_{i\in \mathbb{N}}M_i \to \coprod_{i\in \mathbb{N}}M_i \\
          & (m_i)_{i\in \mathbb{N}} \mapsto (\epsilon_{ij}(m_i) - m_j)_{j \in \mathbb{N}}
\end{align}
and then says that it is clearly a monomorphism.
I'm not quite sure that I understand how this map is defined, neither $i$ nor $j$ are fixed in his definition of the map $\varphi$, so it doesn't seem clear what, say, the third co-ordinate of $\varphi((m_i)_{i\in I})$ is, should it be 
$$\epsilon_{1,3}(m_1) - m_3
$$ or 
$$\epsilon_{56,3}(m_{56})-m_3,
$$
 for example. I think I must be missing something, but I've written out his definition verbatim. 
Also I can't see how this map could be a monomorphism, but maybe that will be clear once I understand how the map is actually defined. 
Thanks very much for any help!
 A: I believe the author is mixing up two results, as it can already be seen with his notation : he starts with an abstract index system $I$, then it becomes $\mathbb{N}$.
Here are some statements which I believe was the author intentions :
let $M_\bullet:I\rightarrow A-Mod$ be any functor. Then we can form a map
$$ \varphi:\coprod_{i\rightarrow j}M_i\rightarrow\coprod_i M_i$$
where the first coproduct is over every map in $I$. This map is defined as follow : recall that a map $ \varphi:\coprod_{i\rightarrow j}M_i\rightarrow\coprod_i M_i$ is a collection of map $\varphi_{i\rightarrow j}:M_i\rightarrow\coprod_j M_j$. This is the universal property of coproduct. Now there is two obvious such maps, namely, the inclusion $\iota_i:M_i\rightarrow\coprod_{j\in I} M_j$, and the composition
$$ M_i\overset{M_{i\rightarrow j}}\longrightarrow M_j\overset{\iota_j}\longrightarrow \coprod_{j\in I}M_j$$
Take the second minus the first and you get your map $\varphi$. Finally the cokernel of this map is indeed the colimit of $F$. Note that this construction is valid in any category, just replace the cokernel of the difference by the coequalizer.
The second statement is quite similar, but it works only with $I=\mathbb{N}$ with its usual order. Now you can indeed define a map
$$\varphi:\coprod_{i\in\mathbb{N}}M_i\rightarrow\coprod_{i\in\mathbb{N}}M_i$$
On $M_i$ it is given by the difference of the usual inclusion $\iota_i:M_i\rightarrow\coprod_{j\in I} M_j$ and the composition 
$$ M_i\longrightarrow M_{i+1}\overset{\iota_{i+1}}\longrightarrow \coprod_{j\in I}M_j$$
So it seems that in the proposition $i$ is just $j-1$.
Finally this last morphism is indeed a monomorphism. Indeed, assume $\varphi(m_i)=0$. The $0$th-coordinate of $\varphi(m_i)$ is $-m_0$. Hence $m_0=0$. The $1$st coordinate is $\epsilon_{01}(m_0)-m_1=-m_1$, so $m_1=0$ and so on.
