Probability measures at play in Bayesian inference

This might be a purely notational, but I'm confused about the probability measures at play when using Bayesian inference. It's sufficient to focus on the numerator here. Let's assume that I have a prior over hypotheses $P(H)$ and that these hypotheses are themselves distributions about some data. When some data $D$ is witnessed, I want to update my prior in the usual way:

$P(H|D) \propto P(D|H) P(H)$. This might be silly, but what I'm confused about is the distribution of the first term, $P(D|H)$. If I'm not completely off track, $P(D|H)$ is $H(E)$, i.e., the probability of the witnessed data under the hypothesis. However, $P(\cdot)$ is a distribution over $H$ and not over $D$, which is why $P(E|H)$ is not making much sense to me.

Where is my thinking wrong? That is, why is $P(E|H)$ defined if $P(\cdot)$ is a distribution over $H$? I'd appreciate any clarifications about the conceptual underpinnings that allow us to go from $P(E|H)$ to $P(H)$ using the same $P(\cdot)$.

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