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, Th\'{e}orie formelle des bicat\'{e}gories

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!