colax map of monads

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 i
diagrammatic order
A ''lax functor P from C to D'', denoted $P: C\to D$, consists of the following data:
* 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, $P_: C(x,y)\to D(P_x,P_y)$;
* for each object ''x∈C'', a 2-morphism $P_:\text_\to P_(\text_x)$ in ''D'';
* for each triple of objects, ''x,y,z ∈C'', a 2-morphism $P_(f,g): P_(f);P_(g)\to P_(f;g)$ 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 $P_$ and $P_{x,y,z}$ above, are invertible is called a pseudofunctor.

Category theory