Why are there only finitely many simplicial maps from one polyhedron to another?

Question

I don't understand why for two polyhedra $|X|$ and $|Y|$, there are finitely many choices of simplicial maps $$s: |X^m| \rightarrow |Y|$$ for some large enough $m \in \mathbb{N}$. Multiple sources say it "follows from the definition," but I don't see how it is that obvious. Is it because $s$ is linear, so it is determined by its effect on vertices, and since there are finitely many vertices in the polyhedron, there are finitely many ways $s$ can affect them? That seems too handwavey. Would anybody care to explain?