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partially ordered category
Let $\mathcal{C}$ be a category in which for every pair of objects $A,B$, the collection $\hom(A,B)$ is a set. $\mathcal{C}$ is called a partially ordered category if $\hom(A,B)\cup\hom(B,A)$ has at most one element. The reason why it is called partially ordered is because we can put a partial order on the objects of $\mathcal{C}$, as follows:
$A\leq B\qquad\mbox{iff}\qquad\hom(A,B)\neq\varnothing.$ 
It is easily verified that $\leq$ is a partial order on $\operatorname{Ob}(\mathcal{C})$: clearly, $\hom(A,A)$ is a singleton as it contains the identity morphism, so that $\leq$ is reflexive; if $\hom(A,B)\neq\varnothing$ and $A\neq B$, then $\hom(B,A)=\varnothing$ by definition, and so $\leq$ is antisymmetric; finally, if neither $\hom(A,B)$ nor $\hom(B,C)$ are empty, then $\hom(A,C)$ can not be empty, as it contains the composition of the elements in $\hom(A,B)$ and $\hom(B,C)$, and thus $\leq$ is transitive.
It is easy to see that any poset can be realized as categroy, and in fact, a partially ordered category.
Given a partially ordered category $\mathcal{C}$, the partial order defined on the objects of $\mathcal{C}$ may be extended to a partial order on the morphisms of $\mathcal{C}$. Formally, let $f,g$ be morphisms in $\mathcal{C}$. Define $f\leq g$ iff $\operatorname{dom}(f)\leq\operatorname{dom}(g)$ and $\operatorname{codom}(f)\leq\operatorname{codom}(g)$. This binary relation can be immediately verified as a partial order (on the class of all morphisms), and an extension of the partial order on objects of $\mathcal{C}$, since every object is identified the corresponding identity morphism. Furthermore, the partial order respects composition, if $f\leq g$, then

$f\circ h\leq g\circ h$, provided that $\operatorname{dom}(f)=\operatorname{dom}(g)=\operatorname{codom}(h)$,

$h\circ f\leq h\circ g$, provided that $\operatorname{codom}(f)=\operatorname{codom}(g)=\operatorname{dom}(h)$.
Combining the two above, we see that $f_{1}\leq g_{1}$ and $f_{2}\leq g_{2}$ imply $f_{1}\circ f_{2}\leq g_{1}\leq g_{2}$, provided that $f_{1}\circ f_{2}$ and $g_{1}\circ g_{2}$ are both defined.
Remarks. From a partially ordered category, one may also define

a linearly ordered category, which is partially category in which $\hom(A,B)\cup\hom(B,A)$ has at least one element for every pair $A,B$ of objects

a directed category, which is a partially ordered category in which for every pair $A,B$ of objects, there is an object $C$ such that $\hom(A,C)$ and $\hom(B,C)$ are both nonempty.

The definition given above of a partially ordered category can be generalized. Instead of having a partial order on the objects of the category, the partial order is now on the morphisms. Let $\mathcal{C}$ be a category such that a partial order $\leq$ is given on the class of morphisms of $\mathcal{C}$. This partial order induces a partial order on the objects: $A\leq B$ iff $1_{A}\leq 1_{B}$. Call $\mathcal{C}$ a partially ordered category if
(a) if $f\leq g$, then $\operatorname{dom}(f)\leq\operatorname{dom}(g)$ and $\operatorname{codom}(f)\leq\operatorname{codom}(g)$,
(b) if $f_{1}\leq g_{1}$ and $f_{2}\leq g_{2}$ such that $f_{1}\circ f_{2}$ and $g_{1}\circ g_{2}$ are defined, then $f_{1}\circ f_{2}\leq g_{1}\circ g_{2}$,
(c) for any $A$ and $f$ such that $A\leq\operatorname{dom}(f)$, there exists a unique $g$ such that $g\leq f$ with $\operatorname{dom}(g)=A$,
(d) for any $A$ and $f$ such that $A\leq\operatorname{codom}(f)$, there exists a unique $g$ such that $g\leq f$ with $\operatorname{codom}(g)=A$.
It’s not hard to see that the above definition does generalize the one given at the beginning of the entry. For example, to see 3., let $f:B\to C$. So $A\leq B$, which means there is a unique $h:A\to B$. Then $g=f\circ h:A\to C$ is the desired morphism: $g$ is unique, $\operatorname{dom}(g)=A$, and $g\leq f$.
References
 1 A. J. Berrick, M. E. Keating, Categories and Modules, with Ktheory in View, Cambridge (2000).
 2 C. Hollins, Extending The EhresmannScheinNambooripad Theorem, (2009)
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