A note on gluing cosilting objects

Based on the recent work of M. Saorin and A. Zvonoreva of gluing (co)silting objects and L.A. Hugel, R. Laking, J. Stovicek and J. Vitora of mutating (co)silting objects, we first study further on gluing pure-injective cosilting objects in algebraic compactly generated triangulated categories and gluing cosilting complexes in the unbounded derived categories of rings. Then we discuss the compatibility of the gluing method and the mutation operation of cosilting objects.
Comment: Correct some errors in Section 5. All comments are very welcome