Let $\mathcal C$ be a category and let $\mathcal F:\mathcal C\to\mathcal C\textrm{at}$ be a strong bifunctor. Given another category $\mathcal D$, let $\triangle_{\mathcal D}$ denote the constant functor $\mathcal C\to\mathcal C\textrm{at}$. Now define lax/strong limits and colimits as follows:

A

**lax limit**of $\mathcal F$ is a category $\mathsf{lim}\mathcal F$ together with a natural equivalence $$[\triangle_{(-)},\mathcal F] \cong \mathcal C\textrm{at}(-,\mathsf{lim}\mathcal F).$$ Here $[\triangle_{(-)},\mathcal F]$ denotes the category of lax natural transformations and modifications.A

**lax colimit**of $\mathcal F$ is a category $\mathsf{colim}\mathcal F$ together with a natural equivalence $$[\mathcal F,\triangle_{(-)}] \cong \mathcal C\textrm{at}(\mathsf{colim}\mathcal F,-).$$- We define
**strong limits**and**strong colimts**by replacing lax natural transormations with strong natural transfomations.

Now, if my calculations are correct, a lax colimit of such a functor $\mathcal F$
is given by the grothendieck construction $\mathcal C\int\mathcal F$ and a lax limit is given by the category of **strict** sections
$s:\mathcal C\to \mathcal C\int\mathcal F$, i.e. the category $\mathcal C\textrm{at}/\mathcal C(\operatorname{id}_\mathcal C,\pi)$, where $\pi:\mathcal C\int\mathcal F\to\mathcal C$ is the opfibration corresponding to $\mathcal F$.

If we consider only the category of opcartesian sections, that is sections that map every morphism in $\mathcal C$ to an opcartesian morphism, we get a strong limit.

Now for the question:

**Is there an explicit description of the strong colimit of a functor $\mathcal F:\mathcal C\to\mathcal C\textrm{at}$?**