Matthew, but what is \$$g\$$ and \$$\lambda\$$ in (2)?

I tried the following, where \$$F\$$ stands for \$$(-)^B\$$:

- let B = 1, then we get \$$X^1 \cong X\$$ and \$$F(f): x \mapsto (f(x)) \$$
- let B = 2, then we get \$$X^2 \cong X \times X\$$ and \$$F(f): (x, x') \mapsto (f(x), f(x'))\$$, or in other words \$$F(f): X \times X \to Y\times Y\$$, which in turn \$$Y \times Y = Y^2\$$
- for arbitrary B (even not finite), we get \$$X^B \cong (X \times X \times ... \times X)_B \$$, and therefore \$$F(f): (x_1, x_2, ...) \mapsto (f(x_1), f(x_2), ...)) \$$

However, at the moment I don't know whether this makes sense and how to make it look better...