A functional S-dual in a strong shape category

In the S-category ${\displaystyle \left\{gotP\right\}}$ (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 ${\displaystyle DX,X=(X,n)\in \left\{gotP\right\}}$, turns out to be of the same weak homotopy type as an appropriately defined functional dual ${\displaystyle \overline{{\left(S^0\right)}^X}}$ (Corollary 4.9). Sometimes the functional object ${\displaystyle \overline{X^Y}}$ is of the same weak homotopy type as the "real" function space ${\displaystyle X^Y}$ (§5).