Let $G$ be a group and $G$-Set the category of left $G$-sets, i.e., sets equipped $X$ with a left action of $G$ on $X$. The forgetful functor from $G$-Set to Set has a right adjoint, which sends each set $S$ to the set of all functions $f:G\to S$, equipped with the $G$-action that sends $(g,f)$ (where $g\in G$ and $f:G\to S$) to the function $(gf):G\to S$ sending any $x\in G$ to $f(xg)$.
Let's say that a monoidal subcategory $Z$ of a monoidal category $(X,\otimes,I)$ is one for which
- Morphisms $f,g\in Z$ imply $f\otimes g\in Z$
- Objects $A,B,C\in Z$ imply the associators $(A\otimes B)\otimes C\to A\otimes(B\otimes C)$ are in $Z$
- $I\in Z$
- 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 $\kappa_Y$.
Best Answer
All that matters is that the hom-set $R(X)(x, y)$ is a singleton for each $x, y \in X$. Identities and composition are therefore trivial, because there's only a single choice. It doesn't matter exactly what the singleton sets are (e.g. whether they are equal or not). You should try to avoid thinking of sets up to equality, and instead consider them only up to isomorphism.