## Co/lax functors without apriori comparison cells

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.

## Morita equivalence by categorical bridges

Let denote the category of 2 objects (name them ) and an inverse pair of arrows.

We define a  bridge  as a category over , i.e. a category equipped with a functor . The full subcategories and are called banks of the bridge, and we write .
The arrows in the -preimage of the two arrows in are referred to as through arrows or heteromorphisms in the bridge.
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