= Solution
A <monoidal category> consists of a <category> $\mathcal C$, a <bifunctor> $\otimes:\mathcal C\times\mathcal C\to\mathcal C$, a <monoidal unit object> $I$, and <natural isomorphisms>
$$
\alpha_{A,B,C}:(A\otimes B)\otimes C\to A\otimes(B\otimes C),\quad
\lambda_A:I\otimes A\to A,\quad\rho_A:A\otimes I\to A.
$$
The <associator> obeys the <pentagon identity for a monoidal category>,
$$
\alpha_{A,B,C\otimes D}\,\alpha_{A\otimes B,C,D}
=(1_A\otimes\alpha_{B,C,D})\,\alpha_{A,B\otimes C,D}\,(\alpha_{A,B,C}\otimes1_D),
$$
and the <unitors> obey the <triangle identity for a monoidal category>,
$$
(1_A\otimes\lambda_B)\,\alpha_{A,I,B}=\rho_A\otimes1_B.
$$
<Composition in a category> here acts from right to left. The first equation compares two paths from $((A\otimes B)\otimes C)\otimes D$ to $A\otimes(B\otimes(C\otimes D))$; the second compares two ways of removing the intervening unit.
It is precisely a <bicategory> with one object $*$. The <category> $\mathcal B(*,*)$ is $\mathcal C$: its objects are the 1-morphisms and its <morphisms> are the 2-morphisms. Horizontal <composition in a category> is $\otimes$, the identity 1-morphism is $I$, and the <associator> and <unitors> give the weak associativity and unit constraints. Vertical <composition in a category> is ordinary <composition in a category> in $\mathcal C$; the interchange law follows because $\otimes$ is a <bifunctor>. The bicategory pentagon and triangle axioms are exactly the two equations above. Conversely the sole hom-category of any one-object <bicategory> gives these data, with the convention that horizontal <composition in a category> uses the indicated ordering of tensor factors.
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