- 'Consider the unit interval $I := [0, 1]$. If there were an exponential $I^I$ in $\CompHaus$, by using $1$ as a test object we can see its underlying set would have to be the set of continuous functions $I \to I$, and the evaluation morphism $I^I \times I \to I$ would have to be $(f, x) \mapsto f(x)$. Now consider the sequence $f_n : I \to I$, $x \mapsto x^n$. By assumption, this would have a convergent subnet $f_{n_j} \to g$ for some directed set $J$ and cofinal map $J \to \IN$. But then, since the evaluation morphism $I^I \times I \to I$ is continuous, this would mean $f_{n_j}(x) \to g(x)$ for each $x \in I$. For $x \in [0, 1)$, this implies that $g(x) = 0$, and for $x = 1$, this implies that $g(x) = 1$, contradicting the fact that $g$ must be continuous.'
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