Warning 5.1.4.5. The notions of covariant and contravariant equivalence are generally distinct from one another. For example, the inclusion map $i_1: \{ 1\} \hookrightarrow \Delta ^1$ is a contravariant equivalence with respect to $\Delta ^1$ (see Proposition 5.1.4.13 below). However, $i_1$ is not a covariant equivalence with respect to $\Delta ^1$: note that $i_1$ is a left fibration, but the restriction map

\[ \pi _0(\operatorname{Fun}_{/\Delta ^1}( \Delta ^1, \{ 1\} ) ) \rightarrow \pi _0( \operatorname{Fun}_{/\Delta ^1}( \{ 1\} , \{ 1\} ) ) \]

is not bijective (the left hand side is empty and the right hand side is not).