On the relative dual of an $S^1$-gerbe over an orbifold

We construct a new effective orbifold $\widehat{\Y}$ with an $S^1$-gerbe $c$ to study an $S^1$-gerbe $\mathfrak{t}$ on a $G$-gerbe $\Y$ over an orbifold $\B$. We view the former as the relative dual, relative to $\B$, of the latter. We show that the two pairs $(\Y, \mathfrak{t})$ and $(\widehat{\Y}, c)$ have isomorphic categories of sheaves, and also the associated twisted groupoid algebras are Morita equivalent. As a corollary, the K-theory and cohomology groups of $(\Y, \mathfrak{t})$ and $(\widehat{\Y}, c)$ are isomorphic.