[NB: Throughout this post, let the subscript $i$ range over the set $\unicode{x1D7DA} \equiv \{0, 1\}$.]
Let $(Y, \leqslant)$ be a poset, and $X\subseteq Y$. Let $\iota_i$ be the canonical inclusions $Y \hookrightarrow (Y \amalg Y)$. Define $Z = (Y \amalg Y)/\sim$, where $\sim$ is the smallest equivalence relation on $Y\amalg Y$ that identifies $\iota_0(x)$ and $\iota_1(x)$, for all $x \in X \subseteq Y$. Finally, define functions $f_i = \pi\;{\scriptstyle \circ}\;\iota_i$, where $\pi$ is the canonical projection $(Y \amalg Y) \to Z = (Y \amalg Y)/\sim$.
I'm looking for a construction of a partial order $\leqslant_Z$ on $Z = (Y \amalg Y)/\sim$ such that both $f_0$ and $f_1$ are order-preserving wrt $\leqslant_Z$.
This is what I have so far:
Since the $f_i$ are injections, the $\leqslant_i$ given by
\leqslant_i \;\;=\;\; \{(f_i(y), f_i(y\,')) \;\;|\;\; (y, y\,' \in Y) \wedge (y \leqslant y\,')\} \;\;\cup\;\; I_Z.
...(where $I_Z$ is the identity on $Z$) are well-defined partial orders on $Z$.
Now, let $T$ be the transitive closure of the relation on $Z$ given by $\leqslant_0 \cup \leqslant_1$. I.e. $T = \bigcap_{V \in \mathscr{T}} V$, where $\mathscr{T} \neq \varnothing$ is the family of transitive relations containing $\leqslant_0 \cup \leqslant_1 $. $T$ is obviously transitive, and it is also reflexive, since $I_Z \subseteq V, \forall \; V \in \mathscr{T}$.
The question therefore reduces to whether $T$ is antisymmetric (i.e. $T\cap T^{-1} = I_Z$).
I'm having a hard time finding a halfway reasonable-looking approach to this last point.