Abstract: Next time, I will finish proving things about sigma and tau; in particular that tau is left adjoint to sigma. They will be used to show that the inclusions of the categories of weak, resp. strong, objects has a left, resp. right, adjoint and then use generalities on adjoints to show that they are equivalent categories. In future lecture(s), at least one and at most two, I will show that the chu category is equivalent to the category of weak objects, and therefore to the category of strong ones. Since the chu category is *-autonomous, so are they. Then I will discuss examples. One is groups. The others are actually all examples of one rather general situation. Let K be a spherically complete field (this was new to me, see: https://en.wikipedia.org/wiki/Spherically_complete_field), which includes all locally compact fields, then there are *-autonomous categories starting with the normed K-spaces. This includes the case that K is discrete.