Reuben wrote:

> Then \$$id \$$ = \$$id \circ id\$$ = \$$(I \otimes id) \circ (id \otimes I) \$$ = \$$(I \circ id) \otimes (id \circ I) \$$ = \$$I \otimes I \$$ = \$$I \$$ . Again, lots of these equalities are clearly wrong.

I'm glad you say that they're clearly wrong.

The most scary thing is that you're tensoring a morphism \$$id\$$ with an object \$$I\$$ - or at least, that's what it looks like, since I said in the lecture that \$$I \in \mathbf{Ob}(\mathcal{C}) \$$ is the unit object for the tensor product, while \$$id \$$ is a standard name for an identity morphism from some object to itself - you're not saying which one. Overall, you seem to be saying the morphism \$$id\$$ is equal to the object \$$I\$$, which can't possibly be true.

There are people who tensor morphisms with objects, but nothing I said in the lecture enouraged doing that, or said what it would mean... so let's only tensor objects with objects, and morphisms with morphisms, okay? That's what the functor

$\otimes \colon \mathcal{C} \times \mathcal{C} \to \mathcal{C}$

lets us do, and this should be enough for our puzzles here.