Sorry for screwing this up in Lecture 26. I'll fix it! As Tobias points out, in a lax monoidal monotone \\(f : X \to Y\\) the stuff involving the multiplication and unit in \\(Y\\) is \\(\le\\) the stuff involving the multiplication and unit in \\(X\\):
\[ f(x) \otimes_Y f(x') \le f(x \otimes_X x') \]
\[ I_X \le f(I_Y) .\]
With oplax monoidal monotones, both inequalities are reverse. Everything important works wll this way, and as far as I know nothing works better if you mix the two kinds of inequalities.