Equivalence relation/Quotient set/Universal property/Fact

Universal property of the quotient set

Let ${}M$ be a set, and let ${}\sim$ denote an equivalence relation on ${}M$, with the quotient set ${}M/\sim$. Let ${}\varphi \colon M\rightarrow W$ be a mapping with ${}\varphi (x)=\varphi (y)$ for all ${}x,y\in M$ with ${}x\sim y$.

Then there exists a uniquely determined mapping

${}{\overline {\varphi }}\colon M/\sim \,\,\rightarrow W$ fulfilling

${}\varphi ={\overline {\varphi }}\circ q$.

Proof, Write another proof