The following question is taken from Arrows, Structures and Functors the categorical imperative by Arbib and Manes, and Theory of Mathematical Structures by Jiří Adamek.
$\color{Green}{Background:}$
[From Arbib and Mane]
$\textbf{(1) Definition:}$ We say that a map $A\xrightarrow{h}B$ is a $\textbf{coequalizer}$ iff there exists a pair $p_1,p_2:R\to A$ of amps such that $h\cdot p_1=h\cdot p_2,$ and such that whenever $A\xrightarrow{h'}B'$ satisfies $h'\cdot p_1=h'\cdot p_2,$ there is a unique map $B\xrightarrow{\psi}B'$ such that
$\psi\cdot h=h'.$ In this situation, we call $h$ the $\textbf{coequalizer of}$ $p_1$ $\textbf{and}$ $p_2,$ and write $h=\text{ceeq}(p_1,p_2).$
$\textbf{(2) Definition:}$ Let $(X,s)$ be a $\textbf{C}-$structure and let $E$ be an equivalence relation on $X.$ Then $E$ is a $\textbf{C-congruence}$ and $X/E$ is a $\textbf{quotient structure of}$ $(X,s)$ if the canonical map
$$\eta_E:(X,s)\to X/E$$
has a co-optimal life $t$ in $\textbf{C}(X/E).$ It is usual to say that $(X/E,t)$ is a $\textbf{quotient structure}$ of $(X,s).$
Here is a more extensive definition for quotient object from Adamek's text:
$\textbf{(3) Definition:}$ Let $(X,\alpha)$ be an object, and let $\sim$ be an equivalence relation on the set $X.$ An object $(X/\sim, \bar{\alpha})$ is called the $\textit{quotient object}$ of $(X,\alpha)$ under $\sim$ provided that
$(1)$ $\varphi:(X,\alpha)\to (X/\sim, \bar{\alpha})$ is a morphism;
$(2)$ for each object $(T,\delta)$ and each map $h:X/\sim\to T$ such that $h\cdot\varphi:(X,\alpha)\to (T,\delta)$ is a morphism, $h:(X/\sim,\bar{\alpha})\to (T,\delta)$ is also a morphism.
$\color{Red}{Questions:}$
From the definition of coequalizer, to construct it, I understand that it requires some sort of quotienting process. I have a few quick questions I hope that the experts at MSE can give some clarifications.
(1) Why does the category of coequalizer requires quotienting process.
(2) How is coequalizer related to the notion of quotient object.
(3) Could coequalizer be an example of a quotient object.