The following definition of Kernel Pair is taken from $\textit{Theory of Categories}$ by Nicolae Popescu and Liliana Popescu
Definition of Kernel Pair
$\begin{array}{ccccccccc} Z & \xrightarrow{u} & X\\ \small {v}\big\downarrow & & \big\downarrow\small {f} & \\ X & \xrightarrow{f} & Y \end{array}$
Let $f:X\rightarrow Y$ be a morphism. An ordered pair $(u,v):Z\rightarrow X$ of morphisms is called a $\textit{kernel pair}$ of $f$ if $f\circ u=f\circ v$, and furthermore if for each ordered pair $(u',v'):Z'\rightarrow X$ of morphisms, such that $f\circ u'=f\circ v',$ there exists a unique morphism $g:Z'\rightarrow Z$ with $u\circ g=u'$ and $v\circ g=v'.$
Definition of Cokernel Pair
$\begin{array}{ccccccccc} Z & \xleftarrow{s} & X\\ \small {r}\big\uparrow & & \big\uparrow\small {p} & \\ X & \xleftarrow{p} & Y \end{array}$
Let $p:Y\rightarrow X$ be a morphism. An ordered pair $(r,s):X\rightarrow Z$ of morphisms is called a $\textit{cokernel pair}$ of $p$ if $r\circ p=s\circ p$, and furthermore if for each ordered pair $(r',s'):X\rightarrow Z'$ of morphisms, such that $r'\circ p=s'\circ p,$ there exists a unique morphism $w:Z\rightarrow Z'$ with $w\circ r=r'$ and $w\circ s=s'.$
Questions: I written out the definition of cokernel pair along with the commutative square diagram. I think it is suppose to be a pushout diagram if I reverse all the arrows in the pullback diagram for kernel pair. I would like to know if the definition that I have written out for cokernel pair is correct.
Thank you in advance.