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When compared with set theory, category theory may seem ontologically extravagant; after all, set theory postulates the existence of only one type of entity, the class, while category theory postulates the existence of at least two types, objects and morphisms. However, it is possible to dispense with objects and deal only with morphisms. The basic idea behind this reduction is the observation that objects appear in a onetoone correspondence with their identity morphisms. Thus to say what a category is, we need only specify the axioms the morphisms must obey; ultimately this will allow us to present a category as a sort of partial algebraic system, with a partial binary operation representing composition of morphisms and two unary operations representing the domain and codomain of a morphism.
Let $M$ be a class. Its members will be called morphisms. Then a category on $M$ consists of the following ingredients:

a partial binary operation $\circ\colon M\times M\to M$, called composition;

a unary operation $S\colon M\to M$, called source or domain; and

a unary operation $T\colon M\to M$, called target or codomain.
The above operations are required to satisfy the following axioms.
1. (Right absorption.) For any morphism $f$,
$SSf=TSf=Sf\text{\quad and\quad}TSf=TTf=Tf.$ 2. (Existence of composite morphisms.) The composite morphism $g\circ f$ is defined if and only if $Tg=Sf$; the morphism $g\circ f$ has source $S(g\circ f)=Sf$ and target $T(g\circ f)=Tf$.
$\xymatrix{\bullet\ar@(ul,dl)[]_{{S(g\circ f)=Sf}}\ar[r]^{f}\ar@{.>}[rd]_{{% \exists!\,g\circ f}}&\bullet\ar[d]^{g}\\ &\bullet\ar@(dr,ur)_{{T(g\circ f)=Tg}}}$ 3. (Existence of identity morphisms.) For any morphism $f$, $Tf\circ f=f\circ Sf=f$.
$\xymatrix{\bullet\ar[r]^{f}\ar[d]_{{Sf}}\ar[rd]^{f}&\bullet\ar[d]^{{Tf}}\\ \bullet\ar[r]_{f}&\bullet}$ 4. (Associativity of composition.) Whenever $h\circ g$ and $g\circ f$ are both defined, then $h\circ(g\circ f)=(h\circ g)\circ f$.
$\xymatrix{\bullet\ar[r]^{f}\ar[rd]_{{g\circ f}}&\bullet\ar[d]^{g}\ar[rd]^{{h% \circ g}}&\\ &\bullet\ar[r]_{h}&\bullet}$
Using this definition, we can define an object to be a morphism in the image of $S$; by the right absorption law, this is equivalent to being in the image of $T$. If we also define $\mathrm{Hom}(A,B)$ to be the collection of morphisms $f$ such that $Sf=A$ and $Tf=B$, then we recover the rest of the ordinary definition of category.
References
 1 P. Freyd, A. Scedrov, Categories, Allegories, NorthHolland, New York (1989).
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