String/$(D-5)$-brane Duality and $S$ Duality as Symmetries of Actions

We realize the string/$(D-5)$-brane duality on the action level between the $T^{10-D}$-compactified heterotic string effective action and the $(D-5)$- brane effective action in $D$ dimensions by managing a Lagrange multiplier field. A dual dictionary is composed to be available for the translation between the elementary or solitonic solutions of the dual pair of actions. In the same way the $S$ duality is also reconstructed on the action level as a double dualization for the $T^6$-compactified heterotic string effective action.