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!