small category- alternative

A (small) category $\mathcal{C}$ consists of a set of objects $C_{0}$ and a set of arrows $C_{1}$ together with the following structure:

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a source map $s:C_{1}\to C_{0}$ assigning an object $s(f)$ to each arrow $f\in C_{1}$ ,
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a target map: $t:C_{1}\to C_{0}$ assigning an object t(f) to each arrow $f\in C_{1}$ ,
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an identity map $1:C_{0}\to C_{1}$ assigning to each object $A$ an arrow $1_{A}$ with

$s(1_{A})=t(1_{A})=A,$
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a composition map $\circ:C_{1}\times C_{1}\to C_{1}$ assigning to each pair of arrows $f, g$ , such that

$s(g)=t(f)$

, a third arrow $g\circ f$ with $s(g\circ f)=s(f)$ and $t(g\circ f)=t(g)$ .
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The composition thus defined “ $\circ$ ” is associative, that is,

$h\circ(g\circ f)=(h\circ g)\circ f$

whenever these compositions make sense.
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the identity map satisfies $f\circ 1_{A}=f$ for any $f$ such that $s(f)=A$ and $1_{A}\circ g=g$ , and any $g$ such that $t(g)=A$ .

References

P. A. Zito. 2008. [arXiv: math.CT]. http://arxiv.org/PS_cache/math/pdf/0509/0509266v1.pdf2- $C^{*}$ -Categories with non-simple units. ,(Preprint).