Spectrum functor converts direct limits to inverse limits

Statement

Category-theoretic statement

The natural map from the spectrum of the direct limit of a directed system of commutative unital rings, to the inverse limit of their spectra, is an isomorphism.

Verbal statement

Suppose $I$ is a directed set and $R_{i}$ is a directed system of commutative unital rings indexed by $I$ . Suppose $R$ is the direct limit of the $R_{i}$ s. Then the natural map:

$S p e c (R) \to lim_{\leftarrow} S p e c (R_{i})$

is an isomorphism.