Create 2-category from a cartesian category

Question

I have a way of constructing (something like) a 2-category from a category with products $\mathcal{C}$.

My question: Is this a correct construction, and if so, does it have a name?

Define $\mathcal{C'}$ to consist of one object. The 1-morphims are objects of $\mathcal{C}$, the identity is the terminal object of $\mathcal{C}$, and $A \circ B$ is $A \times B$. The 2-morphisms $A \to B$ are morphisms $A \to B$ from $\mathcal{C}$. Vertical composition is composition in $\mathcal{C}$ and the horizontal composition of $f$ and $g$ is given by $f \times g$.

Notes:

1. I said "something like" a 2-category because it seems like composition will be defined up to isomorphism, rather than equality.

2. Perhaps this would work with any monoidal category, rather than just a category with products.

3. I want this construction so I can "reverse" the usual relationship between monoid object and monad, and say that a monoid in $\mathcal{C}$ is given by a monad in $\mathcal{C'}$.

Answer 1

Yes, and more generally this works for any monoidal category. In fact a monoidal category is precisely a (weak) 2-category with one object, in the same way that a monoid is precisely a category with one object. (Weak 2-categories are the "correct" notion of 2-category; what you have in mind is a strict 2-category.)

The story continues: recall that a monoid in the category of monoids is a commutative monoid by the Eckmann-Hilton argument, and then things stabilize; taking monoid objects doesn't give you anything new. One can ask what a monoid in the category of monoidal categories is, and this turns out to be a braided monoidal category. Then a monoid in the category of braided monoidal categories is a symmetric monoidal category.

More generally, see the periodic table of higher categories.