Dotted $2$-limits

Marked limits, or Cartesian quasi-limits introduced by Gray, give an alternative approach to $\mathbf{Cat}$-weighted limits in $2$-category theory. This was first established by Street, and we aim to give a new approach to this result using marked codescent objects of marked coherence data which we introduce in this article. We then propose the notion of dotted $2$-limits, which is a natural generalisation of marked limits to the enhanced $2$-categorical setting. We establish that dotted $2$-limits and $\mathscr{F}$-weighted limits both have the same expressive power.