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.



how to prove cartesian closed category with finite coproducts is distributive category?
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!


