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.