The following question is taken from Arrows, Structures and Functors the categorical imperative by Arbib and Manes, "Advanced Modern Algebra" 2nd edition by Rotman, and "Representation theory of the Virasoro Algebra" by Iohara and Koga.
$\color{green}{Background}$
From Iohara and Koga's Representation theory of the Virasoro Algebra.
$\textbf{(1):}$ A pair $(C,p)$ is called the $\textit{cokernel}$ of $f$ if it satisfies
i. $p\circ f=0$ and ii. for any morphism such that $q\circ f=0$ there exists a unique morphism $r$ such that $q=r\circ p.$
In such case, the object $C$ is often denoted by $\text{Coker }f$
From Rotman's Advanced Abstract Algebra 2nd edition.
$\textbf{(2)}$ Definition: Given two morphisms $f:A\rightarrow B$ and $g:A\rightarrow C$ in a category $\textbf{C},$ a $\textbf{solution}$ is an ordered triple $(D,\alpha,\beta)$ making the left-hand diagram commute. A $\textbf{pushout}$ (or $\textit{fibered sum}$ is a solution $(D,\alpha,\beta)$ that is "best" in the following sense: for every solution $(X,\alpha',\beta'),$ there exists a unique morphism $\theta:D\rightarrow X$ making the right-hand diagram commute.
Showing that cokernel is a pullback in the case of $_R\textbf{Mod}.$
Suppose $f:A\rightarrow B$ is a homomorphism in $_R\textbf{Mod},$ then the pullback of the first diagram below is $({coker}f,\pi,0),$ where $\pi:B\rightarrow {coker}f$ is the natural map, and ${coker}f=B/\text{im}f.$
$\textbf{(3)}$ $\textbf{Proposition:}$ Every coequalizer is an epimorphism.
Proof: If $h=\text{coeq}(p_1,p_2)$ and $k_1\circ h=k_2\circ h,$ then $$(k_1\circ k_2)\circ p_1=k_1\circ(h\circ P_2)=k_1\circ(h\circ p_2)=(k_1\circ h)\circ p_2$$ and so there is a unique morphism $\psi$ such that $k_1\circ h=\psi\circ h.$ Thus $k_1=\psi=k_2.$
$\textbf{(4) Proposition:}$ In the category $\textbf{Vect}$
$\textbf{(i)}$ $f$ is an epimorphism $\quad\text{iff}\quad$ $f$ is onto $\quad\text{iff}\quad$ $f$ is a coequalizer
$\textbf{(ii)}$ $f$ is a monomorphism $\quad\text{iff}\quad$ $f$ is one-to-one $\quad\text{iff}\quad$ $f$ is an equalizer
$\textbf{(5)}$ $\textit{A coequalizer is an epimorphism.}$
Proof: If $f:A\to B$ $\textit{is an epimorphism, f is onto.}$ Let $E$ be the equivalence relation $\{(b,b')\mid b-b'\in f(A)\}$ on $B.$ Following a standard if sloppy convention, we write the quotient set as $B/f(A)$ rather than $B/E. B/E$ is easily checked to be a vector space with operations $[b]+[b']=[b+b'], \lambda\cdot[b]=[\lambda\cdot b].$
Define the two linear maps $$t:B\to B/f(A):b\mapsto [b]$$ $$u:B\to B/f(A):b\to [0].$$ Then $t\circ f$ and $u\circ f$ are both zero maps (since $[f(a)]=[0]$ for all $a$ in $A$) and so $t=u$ because $f$ is an epimorphism. Thus for all $b\in B, [b]=[0],b\in f(A),$ and $f$ is onto.
$\color{Blue}{Exercise:}$
Show that the the pushout diagram describes the $\textbf{cokernel}$ of $f$ is the construction $\eta:B\rightarrow B/f(A).$ used in the proof of $\textbf{(2)}$
$\color{Red}{Solution:}$
Solution: We relabel the above diagram as:
$\begin{array}{ccccccccc} X & \xrightarrow{f} & Y\\ \small {}\big\downarrow & & \big\downarrow\small {\eta} & \\ 0 & \xrightarrow{} & C \end{array}$
and we note that $C=\text{Coker}(f)=Y/f(X).$ Also, we let $X=A$ and $B=Y,$ and the linear map $\eta:B\to B/f(A)$ to be relabel as $\eta:Y\to Y/f(X)$ in $\textbf{(5)}.$ So suppose $\eta\cdot f=0.$ Let $C'=\text{Coker}(f)$ so that $\eta':Y\to C'$ is another linear map where $\eta'\cdot f=0,$ and $\eta\cdot f=\eta'\cdot f=0.$ Let $\theta:C\to C'$ be a linear map such that $(\theta\cdot \eta)(b)=\theta(\eta(b))=\theta([b])=\eta'(b)=[b],$ for all $b\in Y.$ If $\theta':C\to C'$ is another linear map where $\theta'\cdot \eta=\theta'\cdot \eta.$ Since $\eta$ is an onto map, hence by $\textbf{(4)(i)},$ $\eta$ is an epimorphism. Therefore $\theta=\theta'$ which shows that $\theta$ is unique.
$\color{Red}{Questions:}$
I would like to know if I did the exercise correctly. I tried to imitate how Rotman did the similar exercise for modules. I have also attached screenshots of the diagrams from Rotman's text to make his proof easier to follow. I gave more details on showing the existence of $\theta:C\to C'.$ I am not sure why Rotman did not included it. Thank you in advance.