In the category of partially ordered sets, the morphisms usually considered are (weakly) increasing maps, that is if $(P_1,\leq_1)$ and $(P_2,\leq_2)$ are partial orders, then a map $f:P_1\to P_2$ is a morphism if $x\leq_1 y$ implies $f(x)\leq_2 f(y)$, for all $x,y\in P_1$.
In model theory and logic, we often consider strong homomorphisms (also called reductions) of relation structures, which in this case would be maps $f:P_1\to P_2$ such that $x\leq_1 y$ if and only if $f(x)\leq_2 f(y)$, for all $x,y\in P_1$. In other words, such maps not only preserve order, but preserve non-order as well.
A simple question: Do such maps have a standard name in the literature on partially ordered sets?
the term is exactly : "order-embedding". Such functions are necessarily injective, btw! More generally, embeddings in model theory both preserve and reflect relations.