YOU SAID:

my teachers pets name is sun, which is also my name, and my teachers name

INTO JAPANESE

先生のペットの名前は私の名前、先生の名前も、日です。

BACK INTO ENGLISH

Teacher's pet name is my name, teacher's name day.

INTO JAPANESE

先生のペットの名前は私の名前、先生の名前の日。

BACK INTO ENGLISH

It's my name, teacher's name, is the name of the teacher's pet.

INTO JAPANESE

それは私の名前、先生の名前、先生のペットの名前です。

BACK INTO ENGLISH

It is my name, teacher's name, the name of the teacher's pet.

INTO JAPANESE

それは私の名前、先生の名前、先生のペットの名前です。

BACK INTO ENGLISH

It is my name, teacher's name, the name of the teacher's pet.