Co/lax functors without apriori comparison cells

A profunctor $\ct F:\ct A\ag\ct B$ is functorial iff every object of $\ct A$ has a reflection in $\ct B$ within $\ct F$ . Fixing (arbitrarily) one reflection arrow for each object $\ct A$ yields a functor $F:\ct A\to\ct B$ such that $F_*\cong \ct F$ .
Dually, $\ct F$ is co-functorial iff every object of $\ct B$ has a coreflection in $\ct A$ , yielding a functor $G:\ct B\to\ct A$ such that $G^*\cong\ct F$ .
If both are satisfied, then $F$ above is left adjoint to $G$ .

This analogy continues in dimension 2 (for bicategories or double categories), yielding the colax functors as double profunctors with the reflection property and the lax functors as double profunctors with the coreflection property. If a double profunctor has both property, it determines a colax/lax adjunction.

Double categories

A double category consists of objects ( $A,B,\dots$ ), vertical arrows ( $u,v,w,\dots$ ), horizontal arrows ( $f,g,h,\dots$ ), and cells ( $\alpha,\beta,\dots$ ). Each arrow is assigned its start and end point, as two (not necessarily distinct) objects, called the ‘domain’ and ‘codomain’. Each cell is assigned its boundary $(f,u,v,g)$ , depicted as

$\nzt fu\alpha vg$

There is a horizontal composition on the horizontal arrows $(f,g)\mapsto fg$ , and on the cells $(\alpha,\beta)\mapsto\alpha\beta$ , and a vertical composition $(u,v)\mapsto u;v=\frac uv$ on the vertical arrows and on the cells $(\alpha,\beta)\mapsto\frac\alpha\beta$ . Two items of similar kind are composable (in a specific order) if their respective boundaries coincides.
The compositions has identity elements: each object determines a horizontal and a vertical identity [denoted by $\overline 1_A$ and $1_A$ , respectively], and each arrow $u:A\downarrow B$ [resp. $f:A\to B$ ] determines an identity cell $\overline 1_u$ with boundary $(\overline 1_A,u,u,\overline 1_B)$ [resp. $1_f:(f,1_A,1_B,f)$ ].
And finally, the compositions are associatve: the vertical compositions are assumed to be strictly associative: $u;v;w = (u;v);w=u;(v;w)$ , while the horizontal compositions may only be weakly associative: up to (coherent) vertical isomorphisms $\ fgh\cong (fg)h\cong f(gh)$ .

Examples:

Any category can be seen as a double category, with the arrows playing the role of both vertical and horizontal arrows, and the cells being the commutative squares.
Any bicategory naturally determines a double category, in which the only vertical arrows are the identities. Thus all things written below are valid as well for (co)lax functors between bicategories.
We can consider the double category $\Bbb{SET}$ of sets, with vertical arrows the functions $f:A\to B$ and with horizontal arrows the binary relations $r\subseteq A\times B$ , having a unique cell with border $~~ \nzt{r}{f}{}{g}{s} ~~$ iff $\forall a\in A,\,b\in B\ (a,b)\in r\implies (a^f,b^g)\in s$ .
In the double category of rings, the objects are the rings, the vertical arrows are ring homomorphisms, the horizontal arrows are the bimodules with composition the tensor product, and a cell $~~ \nzt{{}_AM_B}{u}{\alpha}{v}{{}_RN_S} ~~$ is a map $\alpha:M\to N$ which preserves addition and satisfies $(amb)^\alpha=(a^u)m^\alpha(b^v)$ for all $a\in A,\ b\in B$ .
In the double category of categories the objects are the categories, the vertical arrows are functors, the horizontal arrows are the profunctors, and a cell $~~\nzt{\ct F}U\alpha V{\ct G}~~$ is simply a functor $\alpha:\ct F\to\ct G$ such that $\alpha |_{\ct A}=U$ and $\alpha |_{\ct B}=V$ , where profunctors are regarded as categories (over the arrow), as in our profunctor post.

Let $\Bbb D$ be a double category. Then, as vertical composition (either of vertical morphisms or of cells) is strictly associative, we obtain

a category $\Bbb D^\bullet$ with the same objects as $\Bbb D$ and the vertical arrows as arrows (hence forgetting horizontal arrows and cells)
a category $\Bbb D^\to$ with horizontal arrows as objects and cells with vertical composition as arrows (forgetting only the horizontal compositions)

Accordingly, the notation $a\in Ob\Bbb D^\to$ means that $a$ is a horizontal arrow of $\Bbb D$ .

Double profunctors

Let $\Bbb A$ and $\Bbb B$ be double categories. Then a double profunctor from $\Bbb A$ to $\Bbb B$ can be defined as a double category $\Bbb F$ , disjointly containing (isomorphic copies of) $\Bbb A$ and $\Bbb B$ , such that any further cell has a boundary $(f,u,v,g)$ with $f\in\Bbb A$ and $g\in\Bbb B$ . These additional cells are called ‘heterocells‘ or ‘through cells‘, and $u,v$ are the (necessarily vertical) ‘heteromorphisms‘ or ‘through arrows‘.
(Alternatively, a double profunctor can be viewed as a strict functor $\Bbb F\to I$ where $I$ is the ‘vertical arrow’ double category, containing 2 objects, $0,1$ and a single nonidentity vertical arrow $0\downarrow1$ .)

Recall that an arrow $f:A\to B$ in a category is a reflection arrow to a subcategory $\ct B$ if $B\in Ob\ct B$ and for every $f':A\to B'\in Ob\ct B$ , there is a unique $b:B\to B'\,\in\ct B$ such that $f'=fb$ .

Definition. A double profunctor $\Bbb F:\Bbb A\ag\Bbb B$ has the reflection property, if there are a collection $(r_A)_{A\in Ob\Bbb A}$ of heteromorphisms and a collection $(\ro_a)_{a\in Ob\Bbb A^\to}$ of heterocells, such that
a) each $r_A$ is a reflection arrow from $A$ to $\Bbb B^{\bullet}$ within $\Bbb F^{\bullet}$ ,
b) for every horizontal $a:A\to A'$ , the boundary of $\ro_a$ is of the form $(a,r_A,r_{A'},b)$ for some $b\in Ob\Bbb B^\to$ ,
c) and each $\ro_a$ is a reflection arrow from $a$ to $\Bbb B^\to$ within $\Bbb F^\to$ .

The main observation is that these already induce the following comparison cells:

For any $A\in Ob\Bbb A$ , the horizontal identity cell $\overline 1_{r_A}$ of $r_A$ uniquely factors through $\ro_{\overline 1_A}$ , i.e. there is a cell $\fii_A$ such that $\frac{\ro_{\overline 1_A}}{\fii_A}=\overline 1_{r_A}$ .
By b) it follows that the left and right side of the boundary of $\fii_A$ must be vertical identities, also its bottom is a horizontal identity.
For any consecutive horizontal arrows $a,a_1$ in $\Bbb A$ , the horizontal composition $\ro_a\ro_{a_1}$ uniquely factors through $\ro_{aa_1}$ , yielding a cell $\fii_{a,a_1}$ such that $\frac{\ro_{aa_1}}{\fii_{a,a_1}}=\ro_a\ro_{a_1}$ .
Based on similar considerations as above, if we write $A^F$ for the codomain of $r_A$ and $a^F$ for the bottom of $\ro_a$ , then the boundary of $\fii_{a,a_1}$ is $((aa_1)^F,1,1,a^Fa_1^F)$ .

That is, what this data describes is nothing else but a colax functor.
We can also convert this (assuming choice, of course): it is not hard to construct a double profunctor for any colax functor by adding formal heteroarrows and heterocells and defining their horizontal composition using the comparison cells.

Dually, a double profunctor with the coreflection property is basically a lax functor.

And, if a double profunctor has both the reflection and the coreflection property, then the corresponding colax and lax functors determine a so called colax/lax adjunction.

Examples:

Consider the forgetful functor from unital rings to monoids (which forgets addition), and extend it to their bimodules and biacts. This way we obtain a lax functor $R:\Bbb R{\rm ng}\to \Bbb M{\rm nd}$ . The vertical heteromorphisms in the corresponding double profunctor $\Bbb M{\rm nd}\ag\Bbb R{\rm ng}$ are the monoid morphisms from a monoid to (the underlying multiplicative monoid of) a ring, and the heterocells are the biact morphisms from a biact to a bimodule. More explicitly, each heterocell $\tau$ has a boundary
$\NZT AUBf{}gRMS$
where $A,B$ are monoids, $U$ is an $A$ – $B$ -biact, $R,S$ are unital rings, $M$ is an $R$ – $S$ -bimodule, $f,g$ are vertical heteromorphisms. Then $\tau$ is simply a function $U\to M$ that satisfies $(a\cdot u\cdot b)^\tau\ =\ a^f\cdot u^\tau\cdot b^g$ for all $a\in A,\ b\in B,\ u\in U$ .
We obtain a double profunctor $\Bbb{SET}\ag\Bbb{SET}^{co}$ by binary relations as vertical heteromorphisms. Here a heterocell $\nzt ru{}vs$ has each side as a binary relation, and a single heterocell exists with such boundary iff we get both containment of (left-to-right) compositions $r^\smile\,u\,s \le v$ and $r\,v\,s^\smile \le u$ where superscript $\smile$ means to reverse the relation.
The reflection of an object, i.e. a set $A$ , is its power set $P(A)$ with the $\element$ relation as the reflection arrow, and the reflection of a horizontal arrow, i.e. a relation $r:A-B$ is the relation $P(r):P(A)-P(B)$ between the power sets that holds for a pair $X\subseteq A,\ Y\subseteq B$ iff $\forall (a,b)\in r:\,(a\in X\iff b\in Y)$ .

[1] “Bridges and Profunctors”, PhD dissertation, Bertalan Pécsi