A profunctor is *functorial* iff every object of has a reflection in within . Fixing (arbitrarily) one reflection arrow for each object yields a functor such that .

Dually, is *co-functorial* iff every object of has a coreflection in , yielding a functor such that .

If both are satisfied, then above is left adjoint to .

This analogy continues in dimension 2 (for bicategories or double categories), yielding the *colax* functors as double profunctors with the reflection property and the *lax* functors as double profunctors with the coreflection property. If a double profunctor has both property, it determines a colax/lax adjunction.

Continue reading “Co/lax functors without apriori comparison cells”