Quick question: Let $X$ and $Y$ bet two sets and $\sim$ an equivalence relation on $X$. I was wondering what it means to say that a function $f$: $X\to Y$ 'preserves' $\sim$ in this case. Does it mean that given two equivalent elements of $X$, their images under $f$ are identical? Thats what first comes to mind, but I just don't know. Thanks.
The preservation of equivalence relations under a function normally means
$$x \sim y \iff f(x) \sim f(y) \quad \forall x,y$$ but here $\sim$ is not defined on $Y$, so as far as I know there is no general notion of this, as long as e.g. $X\neq Y$ or $f$ isn't a homomorphism of a certain structure.
EDIT: In the given context I can confirm what you said, the author means $$x \sim y \iff f(x) =f(y).$$
You are considering the functions $f = \pi o \bar f: X \to Y$ where $\pi : X \to X/\sim$ and $\bar f : X/\sim \to Y$. This means that $\bar f$ (and therefore $f$ too) maps a whole equivalence class (the elements of $X/\sim$ are equivalence classes) to the same element in $Y$.