I have just started to study spherical objects with Huybrechts book, and I can not solve this (I suppose simple) exercise:
Let $\mathcal E\in D^b(X)$ be a spherical object, show that $T_\mathcal E \simeq T_{\mathcal E[1]} $, where $T$ is the spherical twist.
I know that $\mathcal E[1]$ is also a spherical object, but I have no idea in how to solve using this.
The spherical twist $T_{\mathcal{E}}$ is given by the Fourier--Mukai functor with kernel $$ Cone(\mathcal{E} \boxtimes \mathcal{E}^\vee \to \mathcal{O}_\Delta) $$ on $X \times X$, where the morphism is the composition of the restriction to the diagonal and the trace map.
If you replace $\mathcal{E}$ by $\mathcal{E}[1]$, the first term will be replaced by $$ \mathcal{E}[1] \boxtimes (\mathcal{E}[1])^\vee \cong \mathcal{E}[1] \boxtimes \mathcal{E}^\vee[-1] \cong \mathcal{E} \boxtimes \mathcal{E}^\vee. $$ This canonical isomorphism is compatible with the restriction to the diagonal and with the trace map, hence the Fourier--Mukai kernels for $T_{\mathcal{E}}$ and $T_{\mathcal{E}[1]}$ are isomorphic, hence the functors are isomorphic as well.