Definition 1.1 (Category). A category $C$ consists of:

• (a)
  a collection $\mathrm{ob}C$ of objects $A,B,C,\dots$.
• (b)
  a collection $\mathrm{mor}C$ of morphisms $f,g,h,\dots$.
• (c)
  two operations $\mathrm{dom}$, $\mathrm{cod}$ from $\mathrm{mor}C$ to $\mathrm{ob}C$: we write $f:A\to B$ for “$f$ is a morphism and $\mathrm{dom}f=A$ and $\mathrm{cod}f=B$”.
• (d)
  an operation from $\mathrm{ob}C$ to $\mathrm{mor}C$ sending $A$ to $1_A:A\to A$.
• (e)
  a partial binary operation $(f,g)\mapsto fg$ on $\mathrm{mor}C$, such that $fg$ is defined if and only if $\mathrm{dom}f=\mathrm{cod}g$, and in this case we have $\mathrm{dom}fg=\mathrm{dom}g$ and $\mathrm{cod}fg=\mathrm{cod}f$.

These are subject to the axioms:

• (f)
  $f{1}_A=f$ and $1_{A}g=g$ when the composites are defined.
• (g)
  $f(gh)=(fg)h$ whenever $fg$ and $gh$ are defined.