Let $\mathcal S, \mathcal T$ be triangulated categories and $R: \mathcal S \to \mathcal T$ be an additive functor that commutes with shift. If $L:\mathcal T \to \mathcal S$ is a left adjoint to $R$, then must $L$ also commute with shift?
(Note that $L$ must be an additive functor Are adjoint functors between additive categories additive? )
Let $X$ be an object of $\mathcal{T}$. There is a sequence of isomorphisms of functors \begin{align} \operatorname{Hom}_{\mathcal{T}}(L(X[n]),?) &\cong\operatorname{Hom}_{\mathcal{S}}(X[n],R?)\\ &\cong\operatorname{Hom}_{\mathcal{S}}(X,(R?)[-n])\\ &\cong\operatorname{Hom}_{\mathcal{S}}(X,R(?[-n])\\ &\cong\operatorname{Hom}_{\mathcal{T}}(LX,?[-n])\\ &\cong\operatorname{Hom}_{\mathcal{T}}((LX)[n],?). \end{align} So by Yoneda's Lemma, $L(X[n])\cong (LX)[n]$.
