Dan noted in a Lecture 26 that John switched the unit conditions for lax and oplax in this Lecture. Here is his post:

> [John](https://forum.azimuthproject.org/profile/17/John%20Baez)
I was having a hard time proving [puzzle 83](https://forum.azimuthproject.org/discussion/2098/lecture-27-chapter-2-adjoints-of-monoidal-monotones#latest)
and I think it's because the unit conditions are the other way around for the lax and oplax monotones, respectively:

> - For the lax monotone we should require:
\$$I_Y \le f(I_X) . \$$

> - For the oplax monotone we should require:
\$$f(I_X) \le_Y I_Y . \$$

> And a nitpick:
> one of these equations was using \$$1\$$ rather than \$$I\$$ to denote the unit;
I've noticed this notation is also mixed at the end of the [next lecture](https://forum.azimuthproject.org/discussion/2098/lecture-27-chapter-2-adjoints-of-monoidal-monotones#latest).

For reference we have

- **Lecture 26** Lax: \$$f(I_X) \le_Y I_Y\$$. Oplax: \$$I_Y \le_Y f(I_X) \$$.

- **Lecture 27** Lax: \$$I_Y \le_Y f(I_X) \$$. Oplax: \$$f(I_X) \le_Y I_Y\$$.

My examples and proofs seem to disagree with each other about what the condition *should* be. Here's what I've tried so far:

**An example showing that the condition Lecture 26 should be true:**

Define \$$f: \mathbb Z \to \mathbb R\$$ by \$$f(x) = x+ 1\$$. It has a right adjoint \$$g: \mathbb R \to \mathbb Z\$$ defined by \$$g(y) = \lfloor y-1 \rfloor\$$. So \$g(0) = -1 \leq 0.\$ Since the super-theorem wants \$$g\$$ to be lax monoidal monotone, the unit condition for lax monoidal monotone should be \$$g(I_Y) \leq_X I_X\$$ which is the condition John listed in Lecture 26.

**A hand-wavy reason why the Lecture 26 condition should be true:**

Again, I'm trying working to define the condition so that the super-theorem is true. If \$$g\$$ is a right adjoint that means it is the approximate inverse of \$$f\$$ from below. A perfect inverse would map \$$I_Y\$$ exactly to \$$I_X\$$ so since \$$g\$$ is "from below" if makes sense that \$$g(I_Y) \leq_X I_X\$$.

**Why it would be nice if the condition in Lecture 27 were true:**

I agree with Dan that the Lecture 26 condition seems much more challenging to prove than the Lecture 27 condition: \$$I_X \leq_X g(I_Y) \$$. If this is the unit condition, then it can be proved in the super-theorem as follows: By hypothesis \$$f\$$ is oplax so by the Lecture 27 oplax unit condition \$$f(I_X) \leq_Y I_Y\$$. Since \$$f\$$ is left adjoint to \$$g\$$ this implies that \$$I_X \leq_X g(I_Y)\$$.

I feel like I've got myself all tangled up! What do you all think?