I am unable to understand the meaning of this example below from paragraph in Group Homomorphism chapter in book Contemporary Abstract Algebra by Gallian:
"When defining a homomorphism from a group in which there are several ways to represent the elements, caution must be exercised to ensure that the correspondence is a function. (The term well-defined is often used in this context.) For example, since $3(x+y)=3x+3y$ in $\mathbb{Z}_6$, one might believe that the correspondence $x+\langle3 \rangle$ $ \to 3x$ from $\mathbb{Z}/\langle 3 \rangle$ to $\mathbb{Z}_6$ is a homomorphism. But it is not a function, since $0+\langle3 \rangle$ $=3+\langle3 \rangle$ in $\mathbb{Z}/\langle3 \rangle$ but $3 \bullet0 \neq 3\bullet3$ in $\mathbb{Z}_6."$