Let $X$ and $Y$ be posets and $f, g : X \to Y$ be monotonic functions. I wish to construct the coequalizer $(Z, \; h : Y \to Z)$ of $f$ and $g$.
My attempt
I first define the relation $R \subseteq Y*Y$ by $y R y' \Leftrightarrow \exists x \in X, \; y = f(x) \;\;\&\;\; y' = g(x)$. Let $\equiv$ be the equivalence relation defined as the reflexive, transitive and symmetric closure of $R$.
For $Z$ I take the equivalence classes of $\equiv$ equipped with the order $[y]_{\equiv} \le [y']_{\equiv}$ iff $y \le y'$.
For $h$ I take the function that maps $y$ to $[y]_{\equiv}$ and I need to show that this function is monotonic.
Questions
Is my construction right? If yes, how to prove the monotonicity of $h$? If no, how to correct it?
More generally, can this construction be extended to DCPOs (Directed Complete Partial Orders)? To algebraic DCPOs? In those cases $f$ and $g$ are Scott-continuous and $h$ must be proved Scott-continuous.