Example 5.1.4.4. Let $f: X \rightarrow Y$ be a morphism of simplicial sets. It follows from Proposition 4.4.2.1 that the following conditions are equivalent:

The morphism $f$ is a covariant equivalence relative to $\Delta ^{0}$.

The morphism $f$ is a contravariant equivalence relative to $\Delta ^{0}$.

The morphism $f$ is a weak homotopy equivalence.

See Proposition 5.1.4.16 for a more general statement.