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”