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bicategory
Lax functors
Definition A.123. Let \(\mathcal{C}\) and \(\mathcal{D}\) be bicategories. \(A\) lax functor \(F: \mathcal{C} \rightarrow \mathcal{D}\) consists of the following data:
- A function \(F: \operatorname{ob}(\mathcal{C}) \rightarrow \mathrm{ob}(\mathcal{D})\).
- For every \(x, y \in \mathrm{ob}(\mathcal{C})\), a functor \(F_{x y}: \mathcal{C}(x, y) \rightarrow \mathcal{D}(F x, F y)\).
- For 1-cells \(f: x \rightarrow y\) and \(g: y \rightarrow z\), natural transformations (i.e., 2-cells) \(F g \circ F f \rightarrow F(g \circ f) .\)
- For 0 -cells \(x \in \mathrm{ob}(\mathcal{C})\), natural transformations (i.e., 2 -cells) \(\mathrm{id}_{F x} \rightarrow F\left(\mathrm{id}_{x}\right)\).
- Associativity and unitality diagrams for the 2 -cells described in the preceding items.
