The De Bruijn–Erdős theorem states that when all finite subgraphs of a graph $G$ can be colored with $n$ colors, the same is true for the whole graph.

There is a natural notion of coloring for hypergraphs which is as follows. Let $H= (V, E)$ be a hypergraph, and let $\kappa\neq \emptyset$ be a cardinal. Then a map $c:V\to\kappa$ is said to be a *(hypergraph) coloring* if the restriction $c\restriction_e : e \to \kappa$ is non-constant whenever $e$ has more than $1$ element.

Is the following statement true?

Let $n>1$ be an integer, and let $H=(V,E)$ be a hypergraph such that for all finite $E_0\subseteq E$, the hypergraph $(V,E_0)$ can be colored with $n$ colors. Then $H$ can be colored with $n$ colors.