Recently, I have been studying functions whose domain and codomain are both quotient sets (i.e. sets of equivalence classes). I discovered how it is important to check that such a 'mapping' is well-defined in such circumstances. I was wondering what would happen if the codomain was the Cartesian product of two quotient sets instead. Can you explain the difference between checking this and checking when the codomain is a quotient set? What is the logic behind these differences? Let me be more explicit:
Let $S$ and $T$ be sets, and $\sim_{S}$, $\sim_{T}$, and $\sim_{S \times T}$ be equivalence relations on $S$, $T$ and $S \times T$, respectively. Show that the 'mapping' $f: \dfrac{S\times T}{\sim_{S \times T}} \to \dfrac{S}{\sim_S} \times \dfrac{T}{\sim_T}$ is well-defined.
I know that strategy used in the case where $f : \dfrac{S}{\sim_{S}} \to \dfrac{T}{\sim_T}$ would be to define a function $F: S \to T$ such that $f(\left[ x\right]) = \left[F(x)\right]$ and prove that if $x \sim_S y$ then $F(x) \sim_T F(x)$. How would one have to adapt the strategy to prove the above? What are the reason for that specific adaption?