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



Alternatively, we can consider the profunctor induced by the span


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


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

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
can even condider 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
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.
Theorem. (Yoneda lemma for simple right extensions.)
Every simple right extension of
is a colimit of simple extensions
. (…..)