Warning 5.1.4.6. In the situation of Definition 5.1.4.1, we will often abuse terminology by referring to the morphism $f: X \rightarrow Y$ in the slice category $(\operatorname{Set_{\Delta }})_{/S}$ as a *covariant equivalence* (*contravariant equivalence*) if it is a covariant equivalence relative to $S$. Beware that there is some danger of confusion, because the notion of covariant equivalence depends on $S$. For example, the inclusion map $\{ 1\} \hookrightarrow \Delta ^1$ is not a covariant equivalence relative to $\Delta ^1$ (Warning 5.1.4.5), but *is* a covariant equivalence relative to the final object $\Delta ^{0} \in \operatorname{Set_{\Delta }}$ (Example 5.1.4.4).

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