may explain the related basics, then step by step explanation on how to understand and understand this problem, please cover all parts, thanks !

2. Let S be a set. and let ~ be an equivalence relation on 5'. Let T = Sfrv be the set of equivalence classes. Recall that we have a map cl: 5' > T that carries an elements 6 S to its equivalence class cl(s) = {5' E S : s' N s}. Thust = cl(s) ifand only ils E 1. Now letX be a set. and let f: S -> X be a map. Assume that for every 3, 5' E S, if: ~ 5', then f(s} : f(s'). Prove that there exists a unique mapng > X such thatf = g o cl