I know that cartesian closed category must have finite products and exponential objects. distributive category must have finite product, finite coproduct and s.t. A*0~0, A*B+A*c~A*(B+C). I think somehow I need to use the property of expoential objects to prove the isomorphism, but not sure how to do it. I would be grateful if someone can help me with that.
If $A\times (-)$ is a left adjoint, it must commute with colimits, and in particular, $\oplus$.
That is, write $B_1\oplus B_2$ as $colim_i B_i$, then $A\times colim_iB_i\cong colim_i(A\times B_i)$, from which the result follows.
Since $A\times (-)$ must preserve colimits, also notice that the initial object is the colimit over the empty diagram, so by preservation of colimits, we win again!