For every object (i.e. -enriched category) of Cat, we have to give a functor . By the cartesian closed structure of the -category Cat, we define these to be the hom-functors which are defined in terms of the monoidal structure and the -enrichment data of by setting to be the composites in .
For every -morphism (i.e. -enriched functor}) , we have to give a natural transformation :
Since we have defined to be the hom-functor , to give a natural transformation is to give a natural transformation . We thus define the -indexed family of morphisms in to be simply the family of morphisms in defining the -enriched functor .
The lax naturality of says that for every -morphism (i.e. a -enriched natural transformation) in Cat, the natural transformations and have to satisfy a compatibility condition with the natural transformations and . Explicitly, the condition is that the composite natural transformation is the same as the composite natural transformation . Unraveling the condition leaves us with the requirement that for every pair of objects of the following diagram in must commute:
But a -enriched natural transformation is by definition a collection of morphisms in such that the above diagram commutes.