Remark 1.4.0.2. Condition $(\ast '')$ is commonly known as the weak Kan extension condition. It was introduced by Boardman and Vogt in [MR0420609], who refer to $\infty $-categories as weak Kan complexes. The theory was developed further by Joyal ([MR1935979] and [joyal]), who refers to $\infty $-categories as quasicategories.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$