Construction of a partial order on a quotient of a coproduct [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.

Suppose that $z\leq w$ and $w\leq z$, this means that there are finite sequences so that $z=z_0\leq z_1\leq ...\leq z_n=w$ and $w=w_0\leq w_1\leq ...\leq w_m=z$ where for all $i< n$  $(z_i,z_{i+1})\in (\leq_0\cup \leq_1\cup I_Z)$ and  for all $j<m$    $(w_j,w_{j+1})\in (\leq_0\cup \leq_1\cup I_Z)$. This implies $z$ equivalent to $w$.
