The following question is taken from $\textit{Handbook of Mathematics}$ by Thierry Vialar,page 608
I consulted the Vialar's handbook and hoping to see what the dual formulation of cokernel in terms of pushout diagram look like. Vialar had the definition of Kernel in terms of pullback diagram and also description of its universal property. For cokernel, he only has the pushout diagram along with the text "In a category with a terminal object $1$. The cokernel of a morphism $f:A\rightarrow B$ is the pushout $\text{cokernel}(f)$ in". For both pullback and pushout diagrams, I appended the maps identities: $K\xrightarrow{} 0\xrightarrow{} B =K\xrightarrow{p} A\xrightarrow{f} B$ and $A\xrightarrow{f} B\xrightarrow{i} \text{coker}(f) =A\xrightarrow{} 1\xrightarrow{} \text{coker}(f).$ I also appended diagrams (1') and (2'), both of which included labeled edges.
Definition of Kernel
In a category with an initial object $0$ and pullbacks, the kernel (denote $\text{ker}(f)$) of a morphism $f:A\rightarrow B$ is the pullback $\text{ker}(f)\rightarrow A$ along $f$ of the unique morphism $0\rightarrow B$
$(1)$$\quad$$\begin{array}{ccccccccc} \text{Ker}(f)=K & \xrightarrow{} & 0\\
\quad\small {p}\big\downarrow & & \big\downarrow\small {} & \\
\quad A & \xrightarrow{f} & B
\end{array}$ $K\xrightarrow{} 0\xrightarrow{} B =K\xrightarrow{p} A\xrightarrow{f} B$ $\quad$ $(1')$$\quad\quad$$\begin{array}{ccccccccc} \text{Ker}(f)=K & \xrightarrow{h_1} & 0\\
\quad\small {p}\big\downarrow & & \big\downarrow\small {h_2} & \\
\quad A & \xrightarrow{f} & B
\end{array}$ $h_2\circ h_1=f\circ p$
This determined the object $\text{ker}(f)$ as the object (unique up to unique isomorphism) for which the following universal property holds: "$\textit{for any object }$ $C$ $\textit{and any morphism }$ $s:C\rightarrow A$ $\textit{such that }$ $f\circ s=0$ $\textit{is the zero morphism, there is a unique morphism }$ $\psi:C\rightarrow \text{ker}(f)$ $\textit{such that }$ $s=p\circ \psi.$
Definition of Cokernel
In a category with a terminal object $1$. The cokernel of a morphism $f:A\rightarrow B$ is the pushout $\text{cokernel}(f)$ in
$(2)$$\quad\quad$$\begin{array}{ccccccccc} A & \xrightarrow{f} & B\\
\quad\small {}\big\downarrow & & \big\downarrow\small {i} & \\
\quad 1 & \xrightarrow{} & \text{coker}(f)
\end{array}\\ $
$A\xrightarrow{f} B\xrightarrow{i} \text{coKer}(f) =A\xrightarrow{} 1\xrightarrow{} \text{coKer}(f)$
$(2')$$\quad\quad$$\begin{array}{ccccccccc} A & \xrightarrow{f} & B\\
\quad\small {z_1}\big\downarrow & & \big\downarrow\small {i} & \\
\quad 1 & \xrightarrow{z_2} & \text{coker}(f)
\end{array}$ $i\circ f=z_2\circ z_1$
This determined the object $\text{coker}(f)$ as the object (unique up to unique isomorphism) for which the following universal property holds: "$\textit{for any object }$ $C'$ $\textit{and any morphism }$ $w:B\rightarrow C'$ $\textit{such that }$ $w\circ f=0$ $\textit{is the zero morphism, there is a unique morphism }$ $\phi:\text{coker}(f)\rightarrow C'$ $\textit{such that }$ $w=\phi\circ i.$
Question: Is the corresponding description of the universal property for cokernel formulated correctly?
Thank you in advance