Let C,D be bicategories. We denote composition in diagrammatic order. A lax functor P from C to D, denoted , consists of the following data:
for each object x in C, an object ;
for each pair of objects x,y ∈ C a functor on morphism-categories, ;
for each object x∈C, a 2-morphism in D;
for each triple of objects, x,y,z ∈C, a 2-morphism in D that is natural in f: x→y and g: y→z.
These must satisfy three commutative diagrams, which record the interaction between left unity, right unity, and associativity between C and D. See http://ncatlab.org/nlab/show/pseudofunctor.
A lax functor in which all of the structure 2-morphisms, i.e. the and above, are invertible is called a pseudofunctor.
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