Remark 3.5.7.22. Let $f: X \rightarrow Y$ and be a morphism of Kan complexes, and let $g: Y \rightarrow Z$ be a homotopy equivalence of Kan complexes. Then $f$ exhibits $Y$ as an $n$-truncation of $X$ if and only if $g \circ f$ exhibits $Z$ as an $n$-truncation of $X$. See Corollaries 3.5.1.28 and 3.5.7.12.
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