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
  
• a composition map ∘ : C₁ × C₁ → C₁ assigning to each pair of arrows f,g , such that
  

  , 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,
  

  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.