Given categories and objects , we let denote the category where we freely add an arrow to the disjoint union of and .

This can be achieved e.g. by taking the colimit of the following -shaped diagram:

where

is the terminal category and

contains two objects, denoted by

and

, and a single nonidentity arrow

, whose image in the colimit will be the freely adjoined arrow

.

Alternatively, we can consider the profunctor induced by the span

(which is just the profunctor composition

).

It’s straightforward to check that .

Based on the construction, for any profunctor , the profunctor morphisms (functors that are identical on both and ) are uniquely determined by where they take the freely added arrow , and that can be any heteromorphism in .

**Theorem. **(*Yoneda lemma for profunctors.*) For any profunctor and objects we have a bijection

natural in both

and

.

It’s also straightforward to verify that, for profunctors , a functor that acts as the identity on both and , naturally corresponds to a natural transformation where are the hom functors of restricted to heteromorphisms (i.e. ), and vice-versa.

This way, involving our previous observation about , we can reformulate the above theorem as, for any functor we have

naturally in

.

That clearly implies both the covariant and the contravariant versions of the original Yoneda lemma when applying it either with or with .

**Presheaves as simple extensions**

A presheaf on a category is a functor , so using we can even consider it as a profunctor , which can be viewed as a *left -module*, or… as a **simple right extension** of by a single object (the object of ).

Similarly, a presheaf on the opposite category can be though of as a *right -module*, or as a **simple left extension**.

**Definition.** A category with object is a *simple right extension* of category by a *new object* , if is a full(y embedded) subcategory of , , and is not the domain of any morphism besides its identity .

**Examples.** For any object we obtain a simple right extension by joining a new object and copying all the arrows as where . This extension category will be denoted by .

Observe that by construction, , so it’s just the *hom functor* seen as a profunctor where is embedded into as the new object .

We can also define the empty and the unique simple right extensions by prescribing that each contains exactly 0 or 1 elements, respectively.

If and are simple right extensions of , with new objects , respectively, then a functor is said to be a *morphism of extensions* whenever it takes to and fixes all objects and arrows in .

**Theorem.** (*Yoneda lemma for simple right extensions.*)

For every simple right extension of , the morphisms of extensions correspond to the new arrows in , naturally in .

**Proof.** Such a morphism is uniquely determined by where it takes the copy of in and that can be any new arrow .

**Theorem.** (*co-Yoneda lemma for simple right extensions.*)

Every simple right extension of is a colimit of simple extensions .

**Proof**. Let a simple right extension of with new object be fixed, and consider the following diagram on the *slice category* : an object of that is a new arrow and the diagram assigns the simple extension to it, whereas to an arrow (with ) it assigns .

Now, we get a cocone over to by defining to be the morphism of extensions that corresponds to , and actually any cocone over , with vertex determines a morphism of extensions by sending any new arrow in to the image of (the copy of) under the corresponding leg map of the cocone.