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.
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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!
answered Jan 22, 2011 at 1:06
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1$\begingroup$ Isn't it easier just to say A x (-) preserves finite coproducts (including empty ones) and leave it at that? $\endgroup$ Commented Jan 22, 2011 at 1:43
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$\begingroup$ Dear Qiaochu: Sure, but I worked it out explicitly for the whimsical unknown(google) above. $\endgroup$ Commented Jan 22, 2011 at 2:02
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$\begingroup$ Thank you for your quick reply:) I have another question, how to prove for all objects in the distributive category, every morphism C~0(intial object) is an isomorphism? $\endgroup$ Commented Jan 22, 2011 at 16:05