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