gn.general topology - Is the realization of a proper map of simplicial spaces proper ?

Let $f:X rightarrow Y$ be a map of $m$-dimensional simplicial spaces (which means that all simplices above dimension $m$ are degenerate). Recall, that $f$ is a natural transformation of functors from $Delta$ to spaces. I want to call such a map proper, if each $f_n:X_nrightarrow Y_n$ is proper.

So the question is, whether $f$ is proper if and only if $|f|$ is proper.

The finite dimensionality is required, as the following example shows:
Take $X$ to be any simplicial space with a finite, positive number of nondegenerate simplices in each dimension. Then the map $f:Xrightarrow pt$ is proper (in the notation from above), but $|X|$ is not compact and hence $|f|$ is not proper.

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