Original sentence: If S is a monoid, then quotient semigroup is a monoid with identity.The mapping  is a semigroup homomorphism, called the quotient map, canonical surjection or projection
Edited sentence: Conversely, the kernel of any semigroup homomorphism is a semigroup congruence.The mapping  is a semigroup homomorphism, called the quotient map, canonical surjection or projection; if S is a monoid then quotient semigroup is a monoid with identity