  Given a marked $\infty$-category $\mathcal{D}^{\dagger}$ (i.e. an
$\infty$-category equipped with a specified collection of morphisms) and a
functor $F: \mathcal{D} \to \mathbb{B}$ with values in an $\infty$-bicategory,
we define $\operatorname{colim}^{\dagger} F$, the marked colimit of $F$. We
provide a definition of weighted colimits in $\infty$-bicategories when the
indexing diagram is an $\infty$-category and show that they can be computed in
terms of marked colimits. In the maximally marked case $\mathcal{D}^{\sharp}$,
our construction retrieves the $\infty$-categorical colimit of $F$ in the
underlying $\infty$-category $\mathcal{B} \subseteq \mathbb{B}$. In the
specific case when $\mathbb{B}=\mathfrak{Cat}_{\infty}$, the
$\infty$-bicategory of $\infty$-categories and $\mathcal{D}^{\flat}$ is
minimally marked, we recover the definition of lax colimit of
Gepner-Haugseng-Nikolaus. We show that a suitable $\infty$-localization of the
associated coCartesian fibration $\operatorname{Un}_{\mathcal{D}}(F)$ computes
$\operatorname{colim}^{\dagger} F$. Our main theorem is a characterization of
those functors of marked $\infty$-categories $f:\mathcal{C}^{\dagger} \to
\mathcal{D}^{\dagger}$ which are marked cofinal. More precisely, we provide
sufficient and necessary criteria for the restriction of diagrams along $f$ to
preserve marked colimits.
