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Morita equivalence by categorical bridges

Let \mathcal Iso denote the category of 2 objects (name them 0,1) and 2 nonidentity arrows, f:0\to 1,\ g:1\to 0 which are inverses of each other.

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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