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 $\mathcal F:\mathcal C\to\mathcal C\textrm{at}$?**



Strong colimits of categories.
Adapted from the papers by Fiore and by Porter that I referred to above:


To form the usual lax colimit of P we take the disjoint union of the $P\_i$ for each $i \in C$ and then adjoin new arrows to represent the action of P: for each $m \colon i \to j$ in C and each $X \in P\_i$ there is an arrow $X \to Pm(X)$. Then we quotient by a congruence that ensures that the assignment of $X \to Pm(X)$ to $X$ is lax natural. This lax transformation is the universal cone.


To form the pseudo colimit we simply make sure each $X \to Pm(X)$ is an isomorphism: adjoin a formal inverse along with it and add the requisite equations to the congruence.


