How does a poset category represent an actual poset?

Question

I am told that, given a partially ordered set (poset) $P$ with relation $\leq$, then we may define the coresponding poset category that represents $P$ by letting the objects of the category be $P$ and the morphisms be \begin{equation} \text{Mor}_{\mathcal C}(a,b) = \begin{cases} * & \text{if } a \leq b \\ \emptyset & \text{otherwise} \end{cases} \end{equation} where $* = *_{ab}$ is an arbitrary element. I could figure out how a poset $P$ gives arise to such a category, but I was not able to figure out that $P, \leq$ defines a poset if we are given such a category. Certainly, if we are given such a category with some relation $\leq$ which we don't know a priori that it's a partial order, I can deduce that for all $a \in P, a \leq a$ (since there is always the identity morphism) and also transitivity, but I was not able to figure out that $\leq$ is antisymmetric. I figured that if $x \leq y$ and $y \leq x$, then the $* = *_{xy}$ in $\text{Mor}(x,y)$ must be an isomorphism, but is there a way to deduce that $*_{xy}$ must be $*_{xx}$? And if we cannot, then what if we let $*_{ab}$ for all $a,b \in P$ be a fixed element, say $*$?