For each pair of $0$-cells $x,y$, a groupoid$B(x,y)$, whose objects are called morphisms or $1$-cells and whose morphisms are called 2-morphisms or $2$-cells;
For each $0$-cell $x$, a distinguished $1$-cell $1_x\colon B(x,x)$ called the identity morphism or identity $1$-cell at $x$;
For each triple of $0$-cells $x,y,z$, a functor${\circ}\colon B(y,z) \times B(x,y) \to B(x,z)$ called horizontal composition;
For each pair of $0$-cells $x,y$, a functor ${-}^{-1}\colon B(y,x) \to B(x,y)$ called the inverse operation;
For each pair of $0$-cells $x,y$, two natural isomorphisms called unitors: $id_{B(x,y)} \circ const_{1_x} \cong id_{B(x,y)} \cong const_{1_y} \circ id_{B(x,y)}\colon B(x,y) \to B(x,y)$;
For each quadruple of $0$-cells $w,x,y,z$, a natural isomorphism called the associator between the two functors from $B_{y,z} \times B_{x,y} \times B_{w,x}$ to $B_{w,z}$ built out of ${\circ}$; and
For each triple of $0$-cells $x,y,z$, two natural isomorphisms called the unit and counit between the two composites of ${-}^{-1}$ and $id_{B(x,y)}$ and the constant functors on the relevant identity morphisms;