Lemma 14.18.3. Let $f : U \to V$ be a morphism of simplicial sets. Suppose that (a) the image of every nondegenerate simplex of $U$ is a nondegenerate simplex of $V$ and (b) no two nondegenerate simplices of $U$ are mapped to the same simplex of $V$. Then $f_ n$ is injective for all $n$. Same holds with “injective” replaced by “surjective” or “bijective”.
Proof. Under hypothesis (a) we see that the map $f$ preserves the disjoint union decompositions of the splitting of Lemma 14.18.2, in other words that we get commutative diagrams
And then (b) clearly shows that the left vertical arrow is injective (resp. surjective, resp. bijective). $\square$
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