Monoidal structure on Set
Let \(I\) be any singleton, say \(\{1\}\) and the monoidal product is the cartesian product.
This means that \(\times\) is a functor:
For any pair of sets in \((S,T) \in Ob(\mathbf{Set}\times\mathbf{Set})\), one obtains a set \(S \times T \in Ob(\mathbf{Set})\).
For any pair of morphisms (functions) one obtains a function \((f\times g)\) which works pointwise: \((f\times g)(s,t):=(f(s),g(t))\) which preserves identities and composition.
The bookkeeping isomorphisms are obvious in Set