alternative definition of small category

Definition 0.1. A (small) category 𝒞 consists of a set of objects C₀ and a set of arrows C₁ together with the following structure:

a source map s : C₁ → C₀ assigning an object s(f) to each arrow f ∈ C₁ ,
a target map: t : C₁ → C₀ assigning an object t(f) to each arrow f ∈ C₁ ,
an identity map 1 : C₀ → C₁ assigning to each object A an arrow 1_A with $s(1 ) = t(1 ) = A, A A$
a composition map ∘ : C₁ × C₁ → C₁ assigning to each pair of arrows f,g , such that $s(g) = t(f)$
, a third arrow g ∘ f with s(g ∘ f) = s(f) and t(g ∘ f) = t(g).
The composition thus defined “∘” is associative, that is, $h ∘ (g ∘ f) = (h ∘ g ) ∘ f$
whenever these compositions make sense.
the identity map satisfies f ∘ 1_A = f for any f such that s(f) = A and 1_A ∘g = g, and any g such that t(g) = A.

P. A. Zito. 2008. [arXiv: math.CT]. 2-C^∗ -Categories with non-simple units. ,(Preprint).

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