A bicategorical version of Masuoka's theorem. Applications to bimodules over functor categories and to firm bimodules

We give a bicategorical version of the main result of A. Masuoka ({Corings and invertible bimodules,} {\em Tsukuba J. Math.} $\mathbf{\text{13}}$ (1989), 353–362) which proposes a non-commutative version of the fact that for a faithfully flat extension of commutative rings $R\subseteq S$, the relative Picard group $Pic(S\mathrm{/}R)$ is isomorphic to the Amitsur 1–cohomology group $H^1(S\mathrm{/}R,U)$ with coefficients in the units functor $U$.