Warning 4.1.1.6. In the statement of Example 4.1.1.5, both hypotheses are necessary: a left fibration of simplicial sets need not be a right fibration and vice versa. For example, the inclusion map $\{ 1 \} \hookrightarrow \Delta ^1$ is a left fibration, but not a right fibration (and therefore not a Kan fibration).

$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$