Reference Request: Lax Ends I've read in a few different places that the standard fact
$\text{Nat}\,(F,G) \cong \int_x \text{Hom}\,(Fx,Gx)$
can be upgraded to
$\textbf{LaxNat}\,(F,G) \cong \oint_x\textbf{Hom}\,(Fx,Gx)$.
where the left hand side is the category of lax natural transformations and modifications, and the right hand side is a lax end.
I am looking for a reference that gives the definition of lax end and proves this equivalence.  I do know of the reference, S. Bozapalides, Théorie formelle des bicatégories, but I can't read French and I also can't find a copy.  If someone can link me to the Bozapalides reference would be great.  Or even better would be if there is a reference in English.  Thanks!
 A: This page says that you may be able to get a copy by emailing Andrée Ehresmann.
I don't know the exact answer to your question, but if you can't find a reference then it may be worth recalling that:


*

*For Cat-valued F and G, $\mathrm{Nat}(F,G) \simeq \{F,G\}$, the limit of G weighted by F,

*ends are $\hom$-weighted limits, and

*there are lax morphism classifiers for 2-functors, meaning that $\mathrm{Lax}(F,G) \simeq \mathrm{Nat}(QF,G)$ for another 2-functor $QF$.


So if you define the lax end $\oint_x T(x,x)$ to be the representative of $\mathrm{Lax}(\hom_K, L(1,T))$, then you get $\oint_x [F x, G x] \simeq \mathrm{Lax}(\hom_K, [F-, G-])$, which is not quite what you want, but it's close.
Hope that helps.
A: A reference is Section 7.1 of Fosco Loregian's very nice book on coends, which treats co/lax co/ends. In particular, see Example 7.1.9 for a proof of the formula
$$\mathrm{Nat}_\mathrm{lax}(F,G)=\int_{A\in\mathcal{C}}\mkern-2.05em\square\mkern+1.0em\mathsf{Hom}_{\mathcal{D}}(F(A),G(A)).$$
Another reference for bicategorical coends is Chapter 2 of Alexander Corner's thesis (PDF) or its arXived version, arXiv:1709.01332 [math.CT].
