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.

In other words, a category is a bridge between categories and if these are disjoint full subcategories of and has no more objects.

Recall that the idempotent completion of a category has all the idempotent arrows of as objects (i.e. those satisfying ), and a morphism of is considered as an arrow in whenever .

**Theorem 1.** Two categories are equivalent if and only if there is a bridge between them where each object has an isomorphic fellow object on the other bank (called equivalence bridge).

**Proof.** Given a bridge with this property, using axiom of choice, we can fix an isomorphism for each object on one bank, and use them to define an equivalence functor.

Given equivalent categories , take an isomorphism of their skeletons , this induces a bridge , which can be extended on both banks.

There is a similar characterization for Morita equivalence of categories.

**Theorem 2.** Two categories are Morita equivalent if and only if there is a bridge between them where each (identity) morphism can be written as a composition of through arrows (called Morita bridge).

Put these together, we arrive to the well known

**Theorem 3.** Two categories are Morita equivalent if and only if their idempotent completions are equivalent.

**Proof.** Consider a Morita bridge and take its idempotent completion . For each object we have for some through arrows . But then is an idempotent in , isomorphic to in . This implies that every , thus also every idempotent on has an isomorphic fellow in , so, by symmetry, is an equivalence bridge.

For the other direction, if is an equivalence bridge, consider its restriction on both banks to . Then any is isomorphic to an idempotent in , say by , and in we have maps that ensures a composition of as heteromorphisms.