 Solution

For a tiny object $A$, the exponential functor $E_A=(-{)}^A$ has a right adjoint, say $R_A$. The defining exponential adjunction gives the natural isomorphismsince $(X^B{)}^A\cong X^{A\times B}$. If $A,B$ are tiny objects, the composite on the right is a left adjoint with right adjoint $R_B{R}_A$. Thus $A\times B$ is tiny. The terminal object is tiny because exponentiation by it is naturally the identity, which is its own right adjoint. Induction, including this empty-product case, proves tiny objects are closed under finite products: