Co/lax functors without apriori comparison cells

A profunctor \ct F:\ct A\ag\ct B is functorial iff every object of \ct A has a reflection in \ct B within \ct F. Fixing (arbitrarily) one reflection arrow for each object \ct A yields a functor F:\ct A\to\ct B such that F_*\cong \ct F.
Dually, \ct F is co-functorial iff every object of \ct B has a coreflection in \ct A, yielding a functor G:\ct B\to\ct A such that G^*\cong\ct F.
If both are satisfied, then F above is left adjoint to G.

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.

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Morita equivalence by categorical bridges

Let \mathcal{ISO} denote the category of 2 objects (name them 0,1) and an inverse pair of arrows.

We define a  bridge  as a category over \mathcal{ISO}, i.e. a category \mathcal H equipped with a functor B:\mathcal H\to\mathcal{ISO}. The full subcategories \mathcal A:=B^{-1}(0) and \mathcal B:=B^{-1}(1) are called banks of the bridge, and we write \mathcal H:\mathcal A \rightleftharpoons \mathcal B.
The arrows in the B-preimage of the two arrows in \mathcal{ISO} are referred to as through arrows or heteromorphisms in the bridge.
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