Let $\thicksim$ be an equivalence relation defined on a set $A.$ Assertion: "The quotient $A/{\thicksim}$ is universal with respect to the property of mapping $A$ to a set in such a way that equivalent elements have the same image." i.e., functions $A \xrightarrow{ \phi} Z$ with $Z$ being any set satisfying $a' \thicksim a'' \Rightarrow \phi(a') =\phi(a'').$ These morphisms are objects, $(\phi,Z)$ of a category. Morphisms $( \phi_1 , Z_1) \rightarrow ( \phi_2 , Z_2)$ are defined as commutative diagrams.
Question: Identify clearly the initial and final objects of this category.
I'm a beginner learning Category theory. I know the definitions of initial and final objects as defined as singletons of the corresponding $Hom$ sets. Is there an easier way to interpret universal property ? I'm finiding in difficult to even intutively think what that means. Any help in solving this problem is much appreciated.