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:

$Spec(R)\to \lim _{\leftarrow }Spec(R_{i})$

is an isomorphism.