Right adjoint of a triangle functor is also unique In general, right adjoint of a functor is unique.
In triangulated categories, this is also true. 
My question is why the natural isomorphism between two 
right adjoints is compatible with the triangle structure.  
 A: Yes, provided the notion of adjunction is the one adapted to triangulated functors between triangulated categories: If $\textsf{F}: {\mathscr S}\to{\mathscr T}$ and $\textsf{G}: {\mathscr T}\to{\mathscr S}$ are triangulated  functors - in particular, isomorphisms of functors $\textsf{F}\Sigma\cong\Sigma\textsf{F}$ and $\textsf{G}\Sigma\cong\Sigma\textsf{G}$ have been fixed - a triangulated adjunction $\textsf{F}\dashv\textsf{G}$ is an adjunction of functors $\textsf{F}\dashv\textsf{G}$ such that both the unit and the counit transformations $\textsf{FG}\Rightarrow\textsf{id}$ and $\textsf{id}\Rightarrow\textsf{GF}$ are natural transformations of triangulated functors, i.e. compatible with the above commutation isomorphisms. 
This is precisely what you obtain when you specialize the general notion of an adjunction in a bicategory to the case of the bicategory of triangulated categories and triangulated functors. In particular, since adjoints are unique up to isomorphism in this general framework, triangulated adjoints are unique up to equivalence of triangulated functors.
