Let $\mathcal{C}$ be a cartesian closed and assume that $0\cong 1$ (its initial object is the same as its terminal object). I want a detalied proof of the answer given here: https://mathoverflow.net/questions/19004/is-the-category-commutative-monoids-cartesian-closed/136480#136480
Let $X,Y \in \mathcal{C}$. I think the way to prove this should be the following:
$$\mathcal{C}(0\times X,Y) \cong \mathcal{C}(0,Y^X) \cong 1_?$$
So I think I need to show that $\mathcal{C}(0\times X,Y) \cong \mathcal{C}(X,Y)$. Am I on the right track? If this isomorphism is true, how can I prove it?
Also, what the subscript of $1_?$ should be?
My knowledge on category theory is limited so I would greatly appreciate an answer as basic as possible.
Hint. Using $X \simeq X \times 1$ and $1 \simeq 0$, you can prove $$ \mathcal C(X,Y) \simeq \dots \simeq \{\ast\}. $$
From there, you can deduce that any object is both initial and final.