**Edit** Originally the question was whether an arbitrary diagram of finite CW complexes can be approximated by a diagram of finite simplicial sets. In view of Tyler's comment, this was almost certainly asking for too much. I restricted the question from arbitrary diagrams to simplicial diagrams.

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It is well-known that for any finite CW complex $K$ one can construct a finite simplicial set whose geometric realization is equivalent to $K$. But the construction is not strictly functorial.

Suppose we have a simplicial object in finite CW complexes. Can one always construct a simplicial object of finite simplicial sets, whose geometric realization is connected to the original diagram by a zig-zag of levelwise weak equivalences?

One may ask a similar question in a more general context. Suppose we have an $\infty$-category $\mathcal C$. Suppose $A$ is a collection of (say compact) objects of $\mathcal C$. Let $CW^f(A)$ be the category of finite cellular objects generated by $A$. Roughly speaking, $CW^f(A)$ consists of objects that can be built as a finite homotopy colimit of objects of $A$. I am leaving the definition of $CW^f(A)$ a little vague - feel free to use any reasonable notion. Let $S_\bullet^f(A)$ be the category of simplicial objects in $\mathcal C$ that in each simplicial degree are a finite sum of elements of $A$, and are degenerate above some dimension. Geometric realization gives a functor $S_\bullet^f(A)\to CW^f(A)$. Now we have an obvious generalization of the question:

Suppose we have a simplicial object in $CW^f(A)$. Can we find a simplicial object in $S_\bullet^f(A)$ whose geometric realization is levelwise (weakly) equivalent to it? If it is not true in general, is there a reasonable sufficient assumption?