Tiny objects are closed under finite products

In a Cartesian closed category, exponentiation satisfies $(-{)}^{A\times B}\cong ((-{)}^B{)}^A$. If the two exponential functors have right adjoints, their composite has the composite of those right adjoints in reverse order. Therefore a product of tiny objects is tiny. The terminal object is tiny since exponentiation by it is the identity, covering the empty product.