In Algebra Chapter $0$, for a fixed set $A$, when talking about "The quotient $A/{\sim}$ is universal with respect to the property of mapping $A$ to a set in such a way that equivalent elements have the same image", he explains that the morphism $π: A → A/{\sim}$ where $a \sim b$ is the equivalence relation defined by $\varphi(a)=\varphi(b)$ is an initial object in the category of functions/morphisms from $A → Z$ where $Z$ is any set. So for any morphism $\varphi: A → Z$, $π$ is such that there is a unique morphism $σ$ in our category of morphisms $(σπ= \varphi)$. I see that $π: A → A/{\sim}$ satisifes the definitions of an initial object, but I don't understand how it is a single object since it depends on $\varphi$. If $\varphi: A → Z$ and $\varphi_2: A → Z$ induce different equivalences $\sim, \sim_2,$ then there is no function $κ$: $A/{\sim} → Z$ such that $κπ=\varphi_2$
For example, the singleton $p$ is final (which is analogous to initial) in set since there is a single constant function from any set to $p$. The morphisms are different but the object is the same.