A category $\C$ consists of

A collection of objects $\C_0$
A collection of arrows $\C_1$
Functions $\dom\colon \C_1\to \C_0$ , $\rng\colon \C_1\to \C_0$ , and an operator (written infix) $\circ\colon \C_1\times \C_1\rightharpoonup \C_1$ .

such that

For all $f,g\in \C_1$ , $g\circ f$ is defined iff $\rng f = \dom g$ .
For all $f,g\in \C_1$ , we have $\dom g\circ f = \dom f$ and $\rng g\circ f = \rng g$ .
For all $f,g,h\in \C_1$ , we have $(h\circ g)\circ f = h\circ (g\circ f)$ .
For every $C\in\C_0$ , there exists $1_C\in\C_1$ such that $\rng 1_C = C$ , $\dom 1_C = C$ , and $1_C\circ f = f$ for all such that $\rng f = C$ , and $f\circ 1_C = f$ for all such that $\dom f = C$ .