Daniel, a homomorphism is a "structure preserving map". In the DB example, this means that mapping from one domain to another preserves facts; for posets, a monotone function (poset homomorphism, if you like) preserves ordering.

As a monotone function, we know that:

x ≤ y implies f(x) ≤ f(y)

If we regard the posets/mfs as categories & functors, then we are mapping not just the values, but the ordering relations. Then we might say:

f(x ≤ y) = f(x) ≤ f(y), which just replaces ⋆ with ≤ from the Wikipedia definition. "The mapped ordering between x and y is equal to the ordering between the mapped x and mapped y"

As a monotone function, we know that:

x ≤ y implies f(x) ≤ f(y)

If we regard the posets/mfs as categories & functors, then we are mapping not just the values, but the ordering relations. Then we might say:

f(x ≤ y) = f(x) ≤ f(y), which just replaces ⋆ with ≤ from the Wikipedia definition. "The mapped ordering between x and y is equal to the ordering between the mapped x and mapped y"