Chern connection component adjoint ($(D_E^{\prime }{)}^{\ast }=-{\ast }_{E^{\ast }}{D}_{E^{\ast }}^{\prime }{\ast }_E$)

For the $(1,0)$ component of a Chern connection, $(D_E^{\prime }{)}^{\ast }=-{\ast }_{E^{\ast }}{D}_{E^{\ast }}^{\prime }{\ast }_E$, where the middle operator is the corresponding component of the dual Chern connection. This is the holomorphic-degree counterpart of the bundle-valued Dolbeault adjoint.