@IvanDiLiberti - Thanks! I think that, if you posted an answer, you could explain your arguments in a more convenient and detailed way. This would also give more visibility to your idea.

@IvanDiLiberti - It looks highly interesting to me!

@EmilJeřábek - Yes, you're perfectly right! I'll edit. Thank you very much!

Let $x,\lambda\in\mathbb R$. If, for some value of $x$, the equality $$\frac{1}{1+\lambda} x^2+2 \frac{2+\lambda}{(1+\lambda)^2}x+\frac{4+3\lambda+\lambda^2}{(1+\lambda)^3}=(x+2)^2$$ holds for all $|\lambda|<1$, it holds for all $\lambda\ne-1$, because a nonzero univariate rational fraction can have only finitely many zeros over a field.

@MattFeller - You may want to take a look at my comment to the post mathoverflow.net/a/211521/461

@MattFeller - About your 2nd comment: I was planning to ask this question next week, unless somebody found a nontrivial example with $C^C\sim C$, but now I think that you should ask the question if you feel like doing it. (I also considered adding it to the present question, but I thought then I wouldn't know if I should accept an answer answering one of the 2 questions.) (An intermediate question would be: are there nontrivial examples of categories $C_1,...,C_n$ with $C_{i+1}=C_i^{C_i}$ for $i=1,...,n-1$ and $C_1\sim C_n$?)

@AndrejBauer - The setting I had in mind was ZFC (possibly with universes), but I'm open to other settings. I'd be very happy if you posted your answer. (I won't promise that I would accept it, even if, I'm sure, it would be correct and highly interesting!)

Thanks! I don't doubt that your proof is correct but I'd like to make sure I understand it, and I'm very slow. Shouldn't $\text{End}(G,G)$ be $\text{End}(G)$? Also it seems to me that you found an example with $\mathcal A$ and $\mathcal B$ not equivalent but $\mathcal A^{\mathcal A}$ and $\mathcal B^{\mathcal B}$ isomorphic (not just equivalent). Am I right?