### A functional S-dual in a strong shape category

In the S-category $P$ (with compact-open strong shape mappings, cf. §1, instead of continuous mappings, and arbitrary finite-dimensional separable metrizable spaces instead of finite polyhedra) there exists according to [1], [2] an S-duality. The S-dual $DX,X=(X,n)\in P$, turns out to be of the same weak homotopy type as an appropriately defined functional dual $\overline{{\left({S}^{0}\right)}^{X}}$ (Corollary 4.9). Sometimes the functional object $\overline{{X}^{Y}}$ is of the same weak homotopy type as the “real” function space ${X}^{Y}$ (§5).