Idea
A 2-dinatural transformation is a categorification of a dinatural transformation. It is more complicated since there are three kinds of opposite 2-category, and also because it could be lax, colax, or pseudo as well as strict.
Definition
Let and be 2-categories and let
be 2-functors. A lax 2-dinatural transformation consists of
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For each , a 1-morphism component
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For each 1-morphism in , a 2-morphism component from the composite
to the composite
(This is a generalization of the hexagon identity for an ordinary dinatural transformation.)
-
For any 2-morphism in , the two composites
and
are equal.
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Each 2-morphism component is the identity .
-
For any composable 1-morphisms , the composite of and (together with two functoriality commutative squares for and ) is equal to .
Examples
If one of and is constant and the other depends only on , we obtain a notion of 2-extranatural transformation.
If and each depend on only one factor in , we obtain a notion of strict 2-natural transformation from a 2-functor of any arbitrary variance to a 2-functor of any other arbitrary variance. For example, if and , then a lax 2-dinatural transformation consists of an -indexed family of -morphisms in , and for each two objects of , an -indexed family of -morphisms , so that for every -morphism , we have the commutative diagram of -morphisms in :
where . is whiskering/horizontal composition. Furthermore, given composable -morphisms , the -morphisms and are related via the formula , which says that the pasting diagram of -morphisms:
reduces to
This sort of transformation appears in the category of V-enriched categories, which is a -category which comes with a unit enriched category and either a lax natural transformation (in the case of a monoidal structure on ), or a lax natural transformation (in the case of a closed structure on ).