Let $I$ be a filtered category and $A:I\longrightarrow Mod-R$ a functor. Then
Every element $a\in \underset{\rightarrow}{\operatorname{Colim}} (A_i)$ is the image of some element $a_i\in A_i$ (for some $i\in I$) under the canonical map $A_i\longrightarrow \underset{\rightarrow}{\operatorname{Colim}}(A_i)$.
For every $i$, the kernel of the canonical map $A_i\longrightarrow \underset{\rightarrow}{\operatorname{Colim}} (A_i)$ is the union of the kernels of the map $\varphi:A_i\longrightarrow A_j$ (where $\varphi:i\longrightarrow j$ is a map in $I$)
There is above lemma in Weibel’s introduction to homological algebra section 6,chapter 2. I can’t understand the proof of Part 2. Mainly after he said
we may assume $i=t$.
I can’t understand him.