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\$$:

$f(x) \otimes_Y f(x') \le f(x \otimes_X x')$

and

$I_X \le f(I_Y) .$

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.