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In this paper we construct the sheaf morphism from the sheaf ofpseudodifferential operators to its symbol class. Since the map is hard toconstruct directly, we realize it with two original ideas as follows. First, tocalculate cohomologies we use the theory of \v{C}ech-Dolbeault cohomologyintroduced 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 toconstruct the sheaf morphism, which is actually an isomorphism of sheaves.

The equivalence of pseudodifferential operators and their symbols via \v{C}ech-Dolbeault cohomology