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precategory
A precategory $\mathcal{B}$ consists of the following
1. 2. a set of elements, called arrows or morphisms, for each ordered pair $(A,B)$ of objects in $\mathcal{B}$, usually written $\hom(A,B)$. For any arrow $f\in\hom(A,B)$, $A$ is called the domain of $f$ and $B$ is the codomain of $f$. It is required that $\hom(A,B)\cap\hom(C,D)=\varnothing$ if $(A,B)\neq(C,D)$.
If $Ob(\mathcal{B})$ is a set, then we say that $\mathcal{B}$ is small. A small precategory is just a directed pseudograph (a digraph allowing multiple edges between pairs of vertices), indeed, for the collection of all arrows in $\mathcal{B}$ is a set, written $Mor(\mathcal{B})$. In addition, there are two functions
$\operatorname{dom},\operatorname{codom}:Mor(\mathcal{B})\to Obj(\mathcal{B})$ 
such that $\operatorname{dom}(f)$ is the domain of $f$ and $\operatorname{codom}(f)$ is the codomain of $f$. Note that both $\operatorname{dom}$ and $\operatorname{codom}$ are welldefined functions because if $f\in\hom(A,B)\cap\hom(C,D)$, then $A=B$ and $C=D$, so that both $\operatorname{dom}$ and $\operatorname{codom}$ map $f$ to unique objects $A$ and $B$ respectively.
With the realization that a precategory is essentially a directed graph, we may use the language of graph theory to define concepts such as paths and loops in a precategory. This will allow us to enlarge any precategory to a category. We will carry out the construction below.
Paths Defined
Let $\mathcal{B}$ be a precategory. A path $p$ (in $\mathcal{B}$) is a finite sequence of arrows $f_{1},\ldots,f_{n}$ such that the codomain of $f_{i}$ is the domain of $f_{{i+1}}$. Note that the definition here does not parallel the one given for a graph (as in graph theory), since we allow vertices (domains and codomains), as well as edges (arrows or morphisms) to coincide. The length of a path $p=(p_{1},\ldots,p_{n})$ is defined to be the nonnegative integer $n$.
Given a path $p=(f_{1},\ldots,f_{n})$, we may set the domain of $p$, written $\operatorname{dom}(p)$, to be $\operatorname{dom}(f_{1})$, and codomain of $p$, written $\operatorname{codom}(p)$, to be $\operatorname{codom}(f_{n})$. A loop is a path $p$ where $\operatorname{dom}(p)=\operatorname{codom}(p)$.
Next, for each ordered pair of objects $(A,B)$ in a precategory $\mathcal{B}$, the collection of paths with with domain $A$ and codomain $B$ is a set, and we denote it by $\operatorname{Hom}(A,B)$.
Composition of Paths Defined
Now, let $f\in\operatorname{Hom}(A,B)$ and $g\in\operatorname{Hom}(B,C)$. So $f=(f_{1},\ldots,f_{n})$ and $g=(g_{1},\ldots,g_{m})$. Since $\operatorname{codom}(f_{n})=B=\operatorname{dom}(g_{1})$, we can “concatenate” the two paths and form a new path
$(f_{1},\ldots,f_{n},g_{1},\ldots,g_{m}),$ 
and we write $g\circ f$ for this new path. It is clear that $g\circ f\in\operatorname{Hom}(A,C)$. It is also easy to see that $\circ$ is a function from $\operatorname{Hom}(A,B)\times\operatorname{Hom}(B,C)$ to $\operatorname{Hom}(A,C)$, if we set $\circ(f,g):=g\circ f$. As the “concatenation” operation is evidently associative, $(h\circ g)\circ f=h\circ(g\circ f)$.
Empty Paths Defined
Finally, for each object $A$ in $Ob(\mathcal{B})$, we can artificially associate an empty path $1_{A}$ with $A$, with the following properties

$1_{A}$ is a path with length $0$

$\operatorname{dom}(1_{A})=\operatorname{codom}(1_{A}):=A$; in other words, $1_{A}\in\operatorname{Hom}(A,A)$

for any $f\in\operatorname{Hom}(A,B)$ and $g\in\operatorname{Hom}(C,A)$, $f\circ 1_{A}:=f$ and $1_{A}\circ g:=g$.
The class of all paths, including every empty path for each object, in $\mathcal{B}$ is written $Path(\mathcal{B})$.
Precategory Enlarged to a Category
So if we start out with a precategory $\mathcal{B}$, we end up with a category $\mathcal{\overline{B}}$ such that
1. $Ob(\mathcal{\overline{B}})=Ob(\mathcal{B})$
2. $Mor(\mathcal{\overline{B}})=Path(\mathcal{B})$, such that

domain and codomain of each morphism are defined to be the domain and codomain of the underlying path

for each ordered pair $(A,B)$ of objects in $\mathcal{\overline{B}}$, the collection of morphisms with domain $A$ and codomain $B$ is a set, and is denoted by $\operatorname{Hom}(A,B)$

for every triple of objects $A,B,C$, a function $\circ$ is defined to be the “concatenation” of a path from $A$ to $B$ and a path from $B$ to $C$

the identity morphism $1_{A}$ each object $A$ is just the empty path associated with $A$.

We may embed $\mathcal{B}$ in $\mathcal{\overline{B}}$ so that $\mathcal{B}$ is just a diagram of $\mathcal{\overline{B}}$. Because of this, $\mathcal{B}$ is also known as a diagram scheme. $\mathcal{\overline{B}}$, also written $F(\mathcal{B})$, is known as the free category freely generated by $\mathcal{B}$.
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