A bicategory is a notion of weak 2-category that arises as a category weakly enriched in categories. That is, instead of having hom sets, between any two objects a bicategory has hom categories such that the enriched category laws hold up to invertible 2-cell rather than strictly.

A bicategory $\mathrm{𝒦︀}$ consists of

1. A type of objects ${\mathrm{𝒦︀}}_0$, or 0-cells
2. For all $𝑥,𝑦:{\mathrm{𝒦︀}}_0$, a category ${\mathrm{𝒦︀}}_1(𝑥,𝑦)$. We may elide the subscript and simply write this as $\mathrm{𝒦︀}(𝑥,𝑦)$. Refer to the objects of $\mathrm{𝒦︀}(𝑥,𝑦)$ as 1-cells between $𝑥$ and $𝑦$, and we may write $𝑓:\mathrm{𝒦︀}(𝑥,𝑦)$ as $𝑓:𝑥\to 𝑦$ or $𝑥\stackrel{𝑓}{\to }𝑦$. For $𝑓,𝑔:\mathrm{𝒦︀}(𝑥,𝑦)$, refer to the morphisms in $\mathrm{𝒦︀}(𝑥,𝑦)$ between $𝑓$ and $𝑔$ as 2-cells and write the morphism $𝛼:(\mathrm{𝒦︀}(𝑥,𝑦))(𝑓,𝑔)$ as $𝛼:𝑓\Rightarrow 𝑔$ or $𝑓\stackrel{𝛼}{\Rightarrow }𝑔$
3. For each $𝑥:{\mathrm{𝒦︀}}_0$, an identity 1-cell $1_{𝑥}:\mathrm{𝒦︀}(𝑥,𝑥)$
4. For all $𝑥,𝑦,𝑧:{\mathrm{𝒦︀}}_0$, a composition functor ${\mathrm{𝒦︀}}_{\star }^{𝑥,𝑦,𝑧}:\mathrm{𝒦︀}(𝑥,𝑦)\times \mathrm{𝒦︀}(𝑦,𝑧)\to \mathrm{𝒦︀}(𝑥,𝑧)$. For 1-cells $𝑓:𝑥\to 𝑦$ and $𝑔:𝑦\to 𝑧$, write their composite as $𝑓\star 𝑔:𝑥\to 𝑧$

5. For all $𝑤,𝑥,𝑦,𝑧:{\mathrm{𝒦︀}}_0$, a natural isomorphism, the associator $𝛼$ between the two composite functors $\mathrm{𝒦︀}(𝑤,𝑥)\times \mathrm{𝒦︀}(𝑥,𝑦)\times \mathrm{𝒦︀}(𝑦,𝑧)\to \mathrm{𝒦︀}(𝑤,𝑧)$ that compose the leftmost, respectively rightmost, pair first:

   $${\mathrm{𝒦︀}}_{\star }^{𝑤,𝑦,𝑧}\circ ({\mathrm{𝒦︀}}_{\star }^{𝑤,𝑥,𝑦}\times \text{id})\Rightarrow {\mathrm{𝒦︀}}_{\star }^{𝑤,𝑥,𝑧}\circ (\text{id}\times {\mathrm{𝒦︀}}_{\star }^{𝑥,𝑦,𝑧})$$

   Its component at 1-cells $𝑓,𝑔,ℎ$ is the invertible 2-cell

   $${𝛼}_{𝑓,𝑔,ℎ}:(𝑓\star 𝑔)\star ℎ\Rightarrow 𝑓\star (𝑔\star ℎ)$$

6. For all $𝑥,𝑦:{\mathrm{𝒦︀}}_0$, natural isomorphisms, the left unitor $𝜆$ and right unitor $𝜌$, each between an endofunctor of $\mathrm{𝒦︀}(𝑥,𝑦)$ and the identity functor:

   $${\mathrm{𝒦︀}}_{\star }^{𝑥,𝑥,𝑦}\circ ⟨1_{𝑥},\text{id}⟩\Rightarrow \text{id}{\mathrm{𝒦︀}}_{\star }^{𝑥,𝑦,𝑦}\circ ⟨\text{id},1_{𝑦}⟩\Rightarrow \text{id}$$

   where $1_{𝑥}$ and $1_{𝑦}$ in the pairings $⟨-,-⟩$ denote the constant functors at the identity 1-cells. The components at a 1-cell $𝑓:\mathrm{𝒦︀}(𝑥,𝑦)$ are the invertible 2-cells

   $${𝜆}_{𝑓}:1_{𝑥}\star 𝑓\Rightarrow 𝑓{𝜌}_{𝑓}:𝑓\star 1_{𝑦}\Rightarrow 𝑓$$

7. such that for all $𝑓:\mathrm{𝒦︀}(𝑥,𝑦)$ and $𝑔:\mathrm{𝒦︀}(𝑦,𝑧)$ the triangle below commutes in $\mathrm{𝒦︀}(𝑥,𝑧)$:

8. and such that for all composable 1-cells $𝑓,𝑔,ℎ,𝑘$ the pentagon below commutes: