Like Eric has said --- the construction trivializes on discrete (co)presheaves.
So, let me try to describe the general case. There are two difficulties: first, your question is about covariant functors, and in case of Grothendieck construction we are used to contravariant functors; second, I am going to describe the construction for more general lax functors, despite the fact that I always forget which functors are lax and which are op-lax :-)
The idea is that since the Gorthendieck construction is a generalisation of the Kleisli construction (i.e. a lax functor $1 \rightarrow \mathbf{Cat}$ is tantamount to a monad in $\mathbf{Cat}$, and the Grothendieck construction over such a lax functor gives the Kleisli resolution for the associated monad), the "dual Grothendieck construction" should be a generalisation of the Eilenberg-Moore construction. Specifically, the dual Grothendieck construction should yield a category consisting of collections of algebras and algebra homomorphisms.
Suppose $\Theta \colon \mathbb{C} \rightarrow \mathbf{Cat}$ is a lax functor. The dual Gothendieck construction of $\Theta$ is the category $\bigwedge \Theta$ defined as follows.
An object in $\bigwedge \Theta$ consists of the data:
- for every object $A \in \mathbb{C}$ an object $\lambda_A \in \Theta(A)$,
- for every morphism $f \colon A \rightarrow B \in \mathbb{C}$ a morphism $\lambda_f \colon \Theta(f)(\lambda_A) \rightarrow \lambda_B$
subject to the following laws:
- (multiplication) if $A \overset{f}\rightarrow B \overset{g}\rightarrow C \in \mathbb{C}$, then: $\lambda_g \circ \Theta(g)(\lambda_f) = \lambda_{f \circ g} \circ \mu_{\lambda_A}$, where $\mu \colon \Theta(g) \circ \Theta(f) \rightarrow \Theta(g \circ f)$ is the transformation from the definition of lax functor; diagrammatically:
$$\require{AMScd}\begin{CD}
\Theta(g)(\Theta(f)(\lambda_A)) @>{\mu_{\lambda_A}}>> \Theta(g \circ f)(\lambda_A)\\
@V{\Theta(g)(\lambda_f)}VV @VV{\lambda_{g \circ f}}V \\
\Theta(g)(\lambda_B) @>{\lambda_g}>> \lambda_C
\end{CD}$$
- (unit) if $A \in \mathbb{C}$, then: $\lambda_{\mathit{id}_A} \circ \eta_{\lambda_A} = \mathit{id}_{\lambda_A}$, where $\eta \colon \mathit{id} \rightarrow \Theta(\mathit{id})$ is another transformation from the definition of lax functor; diagrammatically:
$$\begin{CD}
\lambda_A @>{\eta_{\lambda_A}}>> \Theta(\mathit{id}_A)(\lambda_A)\\
@| @VV{\lambda_{\mathit{id}_A}}V \\
\lambda_A @= \lambda_A
\end{CD}$$
A morphism between defined in the above collections of algebras consists of a collection of morphisms between carriers, such that the structures of the algebras are preserved.
In case $\Theta$ is a discrete strict functor, all morphisms from the definition of an object of $\bigwedge \Theta$ collapse to identities.
[EDIT: connection with the other answer]
First, let me point that Mike's answer relies on an old observation made by Jean Benabou, that lax natural transformations between psuedofunctors $\Phi \rightarrow \Theta \colon \mathbb{C} \rightarrow \mathbf{Cat}$ are tantamount to functors between their Grothendieck constructions:
$$\begin{CD}
\int \Phi @>>> \int \Theta \\
@V{\pi_\Phi}VV @VV{\pi_\Theta}V \\
\mathbb{C} @= \mathbb{C}
\end{CD}$$
Everything else is almost tautological:
$$\hom(1, \mathit{laxLim}(\Theta)) \approx \mathit{laxNat}(\delta_1, \Theta) \approx \hom(\pi_{\delta_1}, \pi_\Theta) \approx \hom(\mathit{id}, \pi_\Theta)$$
where the first isomorphism is the definition of lax limit $\mathit{laxLim}(\Theta)$, the second is Benabou's observation, and the third follows from the fact that the Grothendieck construction over terminal functor yields the identity on the base category (i.e. $\int \delta_1 = \int^{C \in \mathbb{C}} \mathbb{C}/C \times \delta_1(C) \approx \int^{C \in \mathbb{C}} \mathbb{C}/C \times 1 \approx \mathbb{C}/1 \approx \mathbb{C}$).
To link this explanation with my answer, notice that if $\Theta$ is a pseudofunctor, then the laws of unit and multiplication make $\lambda_{(-)}$ a functor to the Grothendieck construction $\int \Theta$ --- morphisms from $X$ to $Y$ over $f \colon A \rightarrow B$ in $\int \Theta$ consist of pairs $\langle f \colon A \rightarrow B, h \colon \Theta(f)(X) \rightarrow Y \rangle$; given another morphism $\langle g \colon B \rightarrow C, k \colon \Theta(g)(Y) \rightarrow Z \rangle$ their composition $k \circ h$ is defined as $k \circ \Theta(g)(h) \circ \mu^{-1}$, and identities in $\int \Theta$ are induced by $\eta^{-1}$.
