Spectrum functor converts direct limits to inverse limits

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.