The aim of this work is to give a new description of the geometry appearing in the multi-specialization along a general family of submanifolds (including some important cases as clean intersection or a simultaneously linearizable family of Lagrangian submanifolds in a cotangent bundle) and to extend the multi-specialization, the main ingredient for the functorial study of asymptotics.In the local model we can construct new sheaves of multi-asymptotically developable functions closely related with asymptotics along a subvariety with a simple singularity such as a cusp.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets