For simplicity, let us consider only a functor out of a small category $\mathcal{C}$ to $Set$,
$$
f:\mathcal{C}\to Set,
$$
The Grothendieck construction produces a category (category of elements) $El(f)$ whose objects are $\sqcup\_{c\in \mathcal{C}} f(c)$. Grothendieck construction provides a universal way to compute the colimits.


My question:


Is there a co-form Grothendieck construction (or Grothendieck nstruction for fun)? The co-form of Grothendieck construction should give us a way to compute the limits. 


By nonsense argument, this might be the (internal) Grothendieck construction for $f^{op}: \mathcal{C}^{op}\to Set^{op}$. But I hardly see what is this more concretely. First, the space of elements is dual to above form, perhaps $\Pi\_{c\in \mathcal{C}} f(c)$. 



What is the co-form of Grothendieck construction?
For any (pseudo)functor $f:C\to \mathrm{Cat}$, its Grothendieck construction $\mathrm{El}(f)$ is, as you have said, its oplax colimit. The *lax limit* of $f$ can also be computed as the category of *sections* of the Grothendieck construction. I.e. its objects are functors $s:C\to \mathrm{El}(f)$ for which the composite $C\xrightarrow{s} \mathrm{El}(f)\to C$ is the identity, and its morphisms are natural transformations lying over the identity of $C$.


