Fiber integration of gerbes and Deligne line bundles

Let $\pi: X \to S$ be a family of smooth projective curves, and let $L$ $M$ pair line bundles on $X$. We show that Deligne's bundle $\langle{L,M}\rangle$ can obtained from the $\mathcal{K}_2$-gerbe $G_{L,M}$ constructed in previous work by authors via an integration along fiber map for gerbes categorifies well known one arising Leray spectral sequence $\pi$. Our construction provides full account biadditivity properties $\langle {L,M}\rangle$. The functorial description low degree maps $\pi$ we develop are independent interest, course provide example their application to Brauer group.