for any morphisms $f_1$, $g_1$, $f_2$, $g_2$ such that the composites above are defined. In other words, a category enriched in Pos, the category of posets.

Compare this to the notion of category internal to Pos.

[[David Roberts]]: I agree: the first definition is of a category enriched in [[Pos]], whereas the reference to source and target maps clearly talks about an internal category. I’ve edited it.

Partially ordered $\dagger$-categories

For a partially ordered $\dagger$-category we can (optionally?) require that

(1)$f^{\dagger} \subseteq g^{\dagger} \Leftrightarrow f \subseteq g$

for any morphisms $f$ and $g$.

The following nomenclature points were raised by one of the previous contruibutors. What do people think? Are they justified or in conflict with existing terminology? Are there examples where the conditions come up naturally?

For a partially ordered $\dagger$-category I will call monovalued morphism such a morphism $f$ that $f \circ f^{\dagger} \subseteq 1_{\mathrm{Dst} f}$.

For a partially ordered category with inverses I will call entirely defined morphism such a morphism $f$ that $f^{\dagger} \circ f \supseteq
1_{\mathrm{Src} f}$.

Partially ordered groupoids

The particular case of an ordered groupoid (in which each $f^{-1}$ is the inverse of $f$) is called an ordered groupoid. This has been studied extensively by Mark Lawson, for instance see

Lawson, Mark V Constructing ordered groupoids Cahiers de Topologie et Géométrie Différentielle Catégoriques, 46 no. 2 (2005), p. 123-138, (numdam)

or his book:

M.V. Lawson, Inverse semigroups: the theory of partial symmetries , World Scientific, 1998.

He makes the point that they were initially studied by Ehresmann and are very closely related to inverse semi-groups.