Lax functor

In category theory, a discipline within mathematics, the notion of lax functor between bicategories generalizes that of functors between categories.

Let C,D be bicategories. We denote composition in diagrammatic order. A lax functor P from C to D, denoted $$P: C\to D$$, consists of the following data: 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.
 * for each object x in C, an object $$P_x\in D$$;
 * for each pair of objects x,y ∈ C a functor on morphism-categories, $$ P_{x,y}: C(x,y)\to D(P_x,P_y)$$;
 * for each object x∈C, a 2-morphism $$P_{\text{id}_x}:\text{id}_{P_x}\to P_{x,x}(\text{id}_x)$$ in D;
 * for each triple of objects, x,y,z ∈C, a 2-morphism $$P_{x,y,z}(f,g): P_{x,y}(f);P_{y,z}(g)\to P_{x,z}(f;g)$$ in D that is natural in f: x→y and g: y→z.

A lax functor in which all of the structure 2-morphisms, i.e. the $$P_{\text{id}_x}$$ and $$P_{x,y,z}$$ above, are invertible is called a pseudofunctor.