  In this paper we construct the sheaf morphism from the sheaf of
pseudodifferential operators to its symbol class. Since the map is hard to
construct directly, we realize it with two original ideas as follows. First, to
calculate cohomologies we use the theory of \v{C}ech-Dolbeault cohomology
introduced by Honda, Izawa and Suwa. Secondly we construct a new symbol class,
which is called the symbols of $C^\infty$-type. These ideas enable us to
construct the sheaf morphism, which is actually an isomorphism of sheaves.
