Suppose that the category $\mathbb C$ is cartesian closed. How do I show that $C^T$ is isomorphic to $C$ for every object $C$, when $T$ is the terminal object?

I'm quite sure that I should solve the exercise rearranging the facts that $C\times T\cong C$ and that the functor $-^T$ preserves limits, being a right adjoint; still, I couldn't come up with any useful idea. I don't want you to solve it, I'd just like a small hint (since this seems an exercise that, once you had the right guess, becomes trivial). Thanks in advance

  • 2
    $\begingroup$ Ah, I misread the question, sorry. $-^T$ is not only a right adjoint but specifically it's the right adjoint of $-\times T$ which is naturally isomorphic to the identity functor. $\endgroup$
    – Berci
    Nov 16, 2021 at 7:54
  • 2
    $\begingroup$ To follow Berci's path you have to use that adjoints are essentially unique: prove that $A\times 1\cong A$ (naturally in A), so that the right adjoint to $-\times 1$, namely exponentiating by 1, is naturally isomorphic to the right adjoint of the identity, namely to the identity. The exercise is done now, but phrased in this way it forces you to fill a lot of details and reason abstractly. $\endgroup$
    – fosco
    Nov 16, 2021 at 8:23
  • $\begingroup$ Thank you I see, I wasn't thinking that adjoints are essentially unique $\endgroup$
    – Dr. Scotti
    Nov 16, 2021 at 8:49


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