Given two sets $X$ and $Y$ and two equivalence relations $\sim_X$ and $\sim_Y$ defined on $X$,$Y$ respectively, then a function (a binary relation which is functional and is left-total) is called morphism if two equivalent arguments $x$ and $y$ on $X$ with respect to $\sim_X$ are mapped to equivalent values on $Y$ with respect to $\sim_Y$.
This definition is so abstract and I cannot deeply understand it.
what are some nice examples of morphisms? (I prefer not to discuss about some abstract algebra concepts if it's possible)