small category- alternative

Definition 0.1.

A (small) category $𝒞$ consists of a set of objects $C_0$ and a set of arrows$C_1$ together with the following structure:

• a source map $s:C_1\to C_0$ assigning an object $s(f)$ to each arrow $f\in C_1$ ,

• a target map: $t:C_1\to C_0$ assigning an object t(f) to each arrow $f\in C_1$ ,

• an identity map $1:C_0\to C_1$ assigning to each object $A$ an arrow $1_A$ with

  $$s(1_A)=t(1_A)=A,$$

• 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 $t(g\circ f)=t(g)$.

• The composition thus defined “$\circ$” is associative, that is,

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

  whenever these compositions make sense.

• 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 $t(g)=A$.