Adjoint functor theorem applied to a forgetful functor

Question

Let $\mathbf{Cat}$ denote the category of small categories and $\mathbf{MCat}$ the category of small monoidal categories with monoidal functors. Consider the forgetful functor $\operatorname{U}:\mathbf{MCat} \rightarrow \mathbf{Cat}$. I want to show that this functor has a left adjoint by using a version of the adjoint functor theorem (and without giving an explicit construction of a left adjoint). I was able to verify that $\mathbf{MCat}$ is locally small, small complete, and that $U$ preserves small limits.

• If I want to apply GAFT, all that remains is to verify the solution set condition. Any ideas on how this can be done?
• If I want to apply SAFT I have to verify that $\mathbf{MCat}$ is well-powered and has a small cogenerating set. Is this true?

This is all self-taught, so that I am still struggling with the individual concepts. So please excuse if the questions are stupid.

Answer 1

Let's say that a monoidal subcategory $Z$ of a monoidal category $(X,\otimes,I)$ is one for which

1. Morphisms $f,g\in Z$ imply $f\otimes g\in Z$
2. Objects $A,B,C\in Z$ imply the associators $(A\otimes B)\otimes C\to A\otimes(B\otimes C)$ are in $Z$
3. $I\in Z$
4. An object $A\in Z$ implies the unitors $I\otimes A\cong A\cong A\otimes I$ are in $Z$.

Evidently the intersection of families of monoidal subcategories is a monoidal subcategory. Therefore, given a functor $F\colon Y\to X$ there is a smallest monoidal subcategory $MF(Y)$ containing the image of $F(Y)$. In particular, $F\colon Y\to X$ factors as $Y\to MF(Y)\hookrightarrow X$. Moreover, $MF(Y)\hookrightarrow X$ is a monoidal functor, i.e. a functor in the category of small monoidal categories.

I now claim the morphisms in this subcategory are exactly the composites of morphisms of the form $F(f_1)\otimes F(f_2)\otimes...\otimes F(f_n)$ (wih various parenthesizations) for $f_i$ morphisms in $X$, and appropriate unitors and associators.

It follows that $MF(Y)$ has cardinality bounded by $\kappa_Y$, where $\kappa_Y$ is the smallest infinite cardinal bounding the cardinality of $Y$.

If $\lambda$ is its cardinal, then the isomorphism $\lambda\cong UM(Y)$ of $\lambda$ with the set of morphisms of $MF(Y)$ induces a monoidal category structure on $\lambda$ so that the resulting monoidal category is isomorphic to $MF(Y)$.

Thus the functor $F\colon Y\to X$ factors as $Y\to M\to X$ where $M\to X$ is a monoidal functor, and $UM$ is a cardinal bounded by $\kappa_Y$. Since the set of cardinals bounded by $\kappa_Y$ is a set, and since each set has a set of monoidal structures, and since between any two categories there is a set of functors between them, it follows that for every category $Y$ there is only a set of functors $Y\to M$ where $M$ is a monoidal category with $UM$ a cardinal bounded by $\kappa_Y$.

By the previous discussion, this is a solution set for the forgetful functor from small monoidal categories to small categories: any functor $F\colon Y\to X$ factors as $Y\to M\to X$ where $M\to X$ is a monoidal functor with $UM$ a cardinal bounded by $\kappa_Y$.