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Let $X$,$Y$, and $Z$ be (compactly generated) spaces. Suppose $f:Z \to Y^X$ is a local homeomorphism. How can we tell this from its adjoint $\tilde f:Z \times X \to Y$? I.e., I want a property $P$ such that $f$ is a local homeomorphism if and only if $\tilde f$ has property $P$ (and I don't want this to be a tautology).

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