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I am working in CCC which has initial objects. Let me write $0$ for an initial object and $1$ for a terminal object. I was able to prove that $A^0 \cong 1$ for any object A. Next I was hoping to prove that $0^A \cong 0$ if $A$ is not isomorphic to 0(not an initial object).

I don't know if this claim is true in any CCC with initial objects or if it requires some other condition on the category. At least it's true in the category of sets.

I tried to obtain an arrow from $0^A$ to arbitrary object X. But I can't see a way because only arrows to exponentials come up in the definition of an exponential. Could anyone show me the proof, or is there some required condition on the category (CCC is not enough)?

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    $\begingroup$ Consider the category Set $\times$ Set and the object $A=(0,1)$. $\endgroup$
    – Andreas Blass
    Commented Sep 9, 2022 at 13:48
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    $\begingroup$ Thanks for your comment. Initial object in Set × Set is (0, 0), and $A = (0, 1)$ is not. However $(0, 0)^(0, 1) = (1, 0)$ is not initial. Set × Set is a CCC, but $0^A = 0$ doesn't hold. So my assumption was incorrect. Is this what you mean? $\endgroup$
    – swan59
    Commented Sep 9, 2022 at 13:58
  • $\begingroup$ @swan59 Yes, that's what I was pointing out. I think this is the simplest counterexample. $\endgroup$
    – Andreas Blass
    Commented Sep 9, 2022 at 15:13
  • $\begingroup$ I got it! Thank you! $\endgroup$
    – swan59
    Commented Sep 11, 2022 at 9:32

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