Sentence Agreement Discussion- Let's Apply

"My sister has two children, so she is busy with them in the mornings." 

In the first clause, the predicate (main verb) is "has", which is third person singular. Who has two children? = "my sister" is a noun used as subject, and it is third person singular. The subject and verb in this sentence agree.

In the second clause, the predicate is "is", which also third person singular. Who is busy? = "she" is a pronoun used as subject, and it is third person singular feminine. Its antecedent is the noun "my sister" in the first clause, which is also third person singular. Therefore, the pronoun and its antecedent agree.