Define an equivalence relation ~ on S×T

Question

Make the projection π : S×T → S a special case of the projection to a quotient.

Specifically, define an equivalence relation ~ on S×T such that the projection π: S×T→(S×T)/∼ coincides with π : S×T → S. 

Answer 1

Define the equivalence relation on S×T: (s,t)∼(s',t') if and only if s=s'. Then S×T/∼={[s]|s∈S}. Then it is easy to see that the given equivalence relation is as desired.