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326 U. Bunke, T. Schick, M. Spitzweck, and A. Thom6.1.10 The sheaf T jX 2 Sh Ab S=X gives rise to the stack BT jX (see 2.3.6). ThenX BT jX ! X is the trivial T-banded gerbe over X. Let H ! X be a gerbe withband T jX .Definition 6.6. A dual of the gerbe H ! X is a pair .H 0 ! X; / of a gerbe H 0 ! Xand an isomorphism of gerbes W H ˝X H 0 ! X BT jX .Every gerbe H ! X with band T jX admits a preferred dual H op ! X which wecall the opposite gerbe. The underlying stack of H op is H , but we change its structureof a T jX -banded gerbe using the inversion automorphism 1 W T jX ! T jX .If .H0 0 ! X; 0/ and .H1 0 ! X; / are two duals, then there exists a uniqueisomorphism class of isomorphisms H0 0 ! H 1 0 of gerbes such that the induced diagramH ˝X H00 H ˝X H100 1X BT jXcan be filled with a two-isomorphism. Note that if H 0 ! X is underlying gerbe of thedual of H ! X, then we have the relation of Dixmier–Douady classesd.H/ D d.H 0 /:Lemma 6.7. Let yH ! X and H ! X be gerbes with band T jX . There is a naturalbijection between the sets of isomorphism classes of isomorphisms yH ! H of gerbesover X and isomorphism classes of isomorphisms of T jX -banded gerbes yH ˝ H op !X BT jX .Proof. An isomorphism yH ! H induces an isomorphism yH ˝X H op ! H ˝X H op !X BT jX . On the other hand, an isomorphism yH ˝X H op ! X BT jX presentsyH ! X as a dual of H op ! X. Since W H ˝X H op ! X BT jX presents H asa dual of H op we have a preferred isomorphism class of isomorphisms yH ! H , too.6.1.11 In view of Lemma 6.7, in Definition 6.2 of a T -duality triple ..E; H /; . yE; yH/;u/we can consider u as an isomorphismOp yH op ˝X p H ! E ByE BT jEB yE :The condition 6.2 2. can be rephrased as follows. After restriction to a point b 2 B wecan find isomorphismsv W T n BT ! H b ; Ov W T n BT ! yH b : (50)