Yet another forms of Yoneda lemma

Given categories $\ct A,\ \ct B$ and objects $a\in Ob\ct A,\ b\in Ob\ct B$ , we let $M_{a,b}:\ct A\ag\ct B$ denote the category where we freely add an arrow $m:a\to b$ to the disjoint union of $\ct A$ and $\ct B$ .
This can be achieved e.g. by taking the colimit of the following $\sf M$ -shaped diagram:

$\dia@R=1pc{{\bf 1}\ar[rd]^0 \ar[dd]_a && {\bf 1} \ar[ld]_1 \ar[dd]^b \\ & {\bf 2} & \\ \ct A && \ct B}$

where $\bf 1$ is the terminal category and $\bf 2$ contains two objects, denoted by $0$ and $1$ , and a single nonidentity arrow $0\to 1$ , whose image in the colimit will be the freely adjoined arrow $m$ .
Alternatively, we can consider the profunctor induced by the span $\ct A\overset{a}\ot 1\overset{b}\to\ct B$ (which is just the profunctor composition $a^*\,b_*:\ct A\ag\ct B$ ).

It’s straightforward to check that $M_{a,b}(x\ig y)=\{\alpha m \beta\mid \alpha:x\to a,\ \beta:b\to y\}\cong \ct A(x\ig a)\times\ct B(b\ig y)$ .

Based on the construction, for any profunctor $\ct F:\ct A\ag\ct B$ , the profunctor morphisms $M_{a,b}\to\ct F$ (functors that are identical on both $\ct A$ and $\ct B$ ) are uniquely determined by where they take the freely added arrow $m$ , and that can be any heteromorphism $a\to b$ in $\ct F$ .

Theorem. (Yoneda lemma for profunctors.) For any profunctor $\ct F:\ct A\ag\ct B$ and objects $a\in Ob\ct A,\ b\in Ob\ct B$ we have a bijection

$\ct Prof(M_{a,b}\ig \ct F)\,\cong\,\ct F(a\ig b)$

natural in both $a$ and $b$ .

It’s also straightforward to verify that, for profunctors $\ct F,\ct G:\ct A\ag\ct B$ , a functor $\ct F\to\ct G$ that acts as the identity on both $\ct A$ and $\ct B$ , naturally corresponds to a natural transformation $F\tto G$ where $F,G$ are the hom functors of $\ct F,\,\ct G$ restricted to heteromorphisms (i.e. $\ct A^{op}\times\ct B\to\ct Set$ ), and vice-versa.

This way, involving our previous observation about $M_{a,b}$ , we can reformulate the above theorem as, for any functor $F:\ct A^{op}\times\ct B\to\ct Set$ we have

$Nat(\ct A(-,a)\times\ct B(b,-)\ig F)\,\cong\, F(a,b)$

naturally in $a,b$ .

That clearly implies both the covariant and the contravariant versions of the original Yoneda lemma when applying it either with $\ct A=1$ or with $\ct B=1$ .

Presheaves as simple extensions

A presheaf on a category $\ct A$ is a functor $\ct A^{op}\to\ct Set$ , so using $\ct A^{op}\cong \ct A^{op}\times 1$ we can even consider it as a profunctor $\ct A\ag 1$ , which can be viewed as a left $\ct A$ -module, or… as a simple right extension of $\ct A$ by a single object (the object of $1$ ).
Similarly, a presheaf on the opposite category $\ct A^{op}$ can be though of as a right $\ct A$ -module, or as a simple left extension.

Definition. A category $\ct B$ with object $b\in\ct B$ is a simple right extension of category $\ct A$ by a new object $b$ , if $\ct A$ is a full(y embedded) subcategory of $\ct B$ , $\ b\notin Ob\ct A$ , $\ Ob\ct B=Ob\ct A\cup\{b\}\,$ and $b$ is not the domain of any morphism besides its identity $1_b$ .
Examples. For any object $a\in Ob\ct A$ we obtain a simple right extension by joining a new object $b$ and copying all the arrows $\alpha:x\to a$ as $\alpha_b:x\to b$ where $x\in Ob\ct A$ . This extension category will be denoted by $\ct A\>a$ .
Observe that by construction, $\ct A\>a(x\ig b)\cong\ct A(x\ig a)$ , so it’s just the hom functor $\ct A(-\ig a):\ct A^{op}\to\ct Set$ seen as a profunctor $\ct A\ag 1$ where $1$ is embedded into $\ct A\>a$ as the new object $b$ .

We can also define the empty and the unique simple right extensions by prescribing that each $\ct B(a\ig b)$ contains exactly 0 or 1 elements, respectively.

If $\ct B$ and $\ct C$ are simple right extensions of $\ct A$ , with new objects $b,\, c$ , respectively, then a functor $\ct B\to\ct C$ is said to be a morphism of extensions whenever it takes $b$ to $c$ and fixes all objects and arrows in $\ct A$ .

Theorem. (Yoneda lemma for simple right extensions.)
For every simple right extension $\ct B$ of $\ct A$ , the morphisms of extensions $\,(\ct A\>a)\to \ct B\,$ correspond to the new arrows $a\to b$ in $B$ , naturally in $a\in\, Ob A$ .

Proof. Such a morphism is uniquely determined by where it takes the copy of $1_a:a\to a$ in $(\ct A\>a)$ and that can be any new arrow $a\to b$ .

Theorem. (co-Yoneda lemma for simple right extensions.)
Every simple right extension $\ct B$ of $\ct A$ is a colimit of simple extensions $\ct A\>a$ .

Proof. Let a simple right extension $\ct B$ of $\ct A$ with new object $b$ be fixed, and consider the following diagram $D$ on the slice category $\ct A / b$ : an object of that is a new arrow $f:a\to b$ and the diagram assigns the simple extension $\ct A\>a$ to it, whereas to an arrow $\alpha:f\to g$ (with $\alpha:a\to a_1\in\ct A,\ f\,=\,\alpha g$ ) it assigns $\fii\mapsto \fii\alpha$ .

Now, we get a cocone over $D$ to $\ct B$ by defining $f^D\to\ct B$ to be the morphism of extensions $(\ct A\>a)\to\ct B$ that corresponds to $f:a\to b$ , and actually any cocone over $D$ , with vertex $\ct C$ determines a morphism of extensions $\ct B\to\ct C$ by sending any new arrow $f:a\to b$ in $\ct B$ to the image of (the copy of) $1_a$ under the corresponding leg map $(\ct A\>a)\to\ct C$ of the cocone.