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