Mon is the category of moniods, which can be seen as categories with one object. Ord is the category of preorders, which can be seen as categories with up to one morphism in each homset.
Is there some sort of categorical construct that combines Mon and Ord to get Cat the category of all categories (small if you wish.) We can see them as concrete categories if needed.
An observation that's too long for a comment:
Categories are monads in the bicategory $\mathsf{Span}$ of spans. Monoids are monads in the one-object bicategory associated to the monoidal category $(\mathsf{Set},\times)$. Posets are monads in the bicategory $\mathsf{Rel}$ of relations. The correct notion of morphism can in each case be recovered by beefing up the bicategory to a proarrow equipment.
So another version of the question would be whether there's a construction that takes a bicategory and a monoidal category and gives you another bicategory, which sends $(\mathsf{Rel},\mathsf{Set})$ to $\mathsf{Span}$. I'm not sure this is any more promising, but it moves the complexity around a bit.
