[001F] Definition 1·a.

A meta-category $\mathfrak{C}$ is defined by explaining what an object of $\mathfrak{E}$ is, and, given two objects $x,y\in \mathfrak{E}$, what a morphism from $x$ to $y$ is, together with the following operations:

1. for each object $x\in \mathfrak{E}$, an identity map $\Idn{x} : x \to x$,
2. for any two maps $f:x\to y$ and $g:y\to z$, a composite map $f;g : x \to z$,
3. such that the following equations hold: $\Idn{x};h = h\qquad h;\Idn{y} = h\qquad f;(g;h) = (f;g);{h}$