The $\infty$-category of spaces is known to be the $\infty$-category obtained from the (ordinary) category of finite sets by freely adding sifted colimits. (See e.g. Cesnavicius-Scholze https://arxiv.org/abs/1912.10932 §5.1 for a review of this notion and for pointers to Lurie's HTT where this is proven.)

Can this characterization be used (ideally without referring to the model of quasi-categories) to show other properties, such as:

• that colimits in spaces are universal (proven by Lurie in HTT Lemma 6.1.3.14)?
• possibly even that $Cat_\infty$ is compactly generated by $*$ and $\Delta^1$?

The $\infty$-category of spaces has the following properties:

1. It is the $\infty$-category obtained from the (ordinary) category of finite sets by freely adding sifted colimits. (See e.g. Cesnavicius-Scholze https://arxiv.org/abs/1912.10932 §5.1 for a review of this notion and for pointers to Lurie's HTT where this is proven.)
2. (As Tim Campion points out in a comment, another characterization, also in HTT): Spaces are obtained by freely adding arbitrary colimits to the category $\{*\}$.

Can either of these characterizations be used (ideally without referring to the model of quasi-categories) to show other properties, such as:

• that colimits in spaces are universal (proven by Lurie in HTT Lemma 6.1.3.14)?
• possibly even that $Cat_\infty$ is generated by (the compact objects) $*$ and $\Delta^1$?