**premises:**

Let $Clv(\mathcal{A})$ the category with objects the fibrations on $P:\mathcal{C}\to \mathcal{A}$ with a fixed clivege, and with cartesian functors $T: \mathcal{C_1}\to \mathcal{C_2}$ on $ \mathcal{A}$ (i.e. $P_2\circ T=P_1$).

Let $Spl(\mathcal{A})\subset Clv(\mathcal{A})$ the subcategory of split clivages, and clivage preserving functors on $ \mathcal{A}$.

We have a equivalence of $Clv(\mathcal{A})$ with the category

$p$-$Fun(\mathcal{A}^{op})$ of pseudo-functors $\textbf{C}: \mathcal{A}^{op}\to CAT$ and pseudo-transformations

and a equivalence of $Spl(\mathcal{A})$ with the category

$2$-$Fun(\mathcal{A}^{op})$ of 2-functors $\textbf{C}: \mathcal{A}^{op}\to CAT$ and 2-transformations.

For $F: \mathcal{A}\to \mathcal{B}$ we have the pullback functor $F^\ast: Clv(\mathcal{A})\to Clv(\mathcal{B})$ and its restriction $F^\ast: Spl(\mathcal{A})\to Spl(\mathcal{B})$

this latter correspond (by the equivalences above) to the (right) composition by $F^{op}$:

$F_\ast: 2$-$Fun(\mathcal{B}^{op})\to 2$-$Fun(\mathcal{A}^{op})$

and this has a left-adjoint and a right-adjoint give by the respective Kan-extensions (chose $\mathcal{A}$ and $\mathcal{B}$ small or ample the universe) then also $F^\ast$ above as a left adjoint and a right adjoint.

Let $F_\bullet: Clv(\mathcal{B})\to Clv(\mathcal{A})$ the left adjoint

in the equivalent form $F_\bullet: Spl(\mathcal{B})\to Spl(\mathcal{A})$ it is give by:

$F_\bullet(P)(B)= \varinjlim P\circ (\pi^B_F)^{op}= \varinjlim_{a: B\to F(A)} P(A)$

with $\pi^B_F: B\downarrow F\to \mathcal{A}$ natural.

Let $Oub_F: Spl(\mathcal{B})\xrightarrow{F^\ast} Spl(\mathcal{A})\xrightarrow{U} Clv(\mathcal{A})$

($U$ is the trivial forgetful inclusion).

From [G] 2.4.2.1( p.38) we have a left adjoint (is a 2-adjoint):

$Gau_F$ of $Oub_F$, that in term of pseudo.funtors, for $P\in p$-$Fun(\mathcal{A}^{op})$ this is give by:

$Gau_F(P)(B)= \underrightarrow{LIM}\ P\circ (\pi^B_F)^{op}$

where $\underrightarrow{LIM}$ is the pseudo-colimit operator.

Now, for $F=1_\mathcal{A}$ we get a left adjoint $L$ of the inclusion $U: Spl(\mathcal{A}\to Clv(\mathcal{A})$:

$L(P)(A)= \underrightarrow{LIM}\ P\circ (\pi_A)^{op}$ with $\pi_A: A\downarrow \mathcal{A}\to \mathcal{A}$.\

But then $Gau_F\cong F_\bullet\circ L$

and then $\underrightarrow{LIM}\ P\circ \pi^B_F\ = \ \varinjlim_{(a, A)\in B\downarrow F} \underrightarrow{LIM}(P\circ \pi_A)$\

Now, in the category $CAT\downarrow \mathcal{A}$ we have that $\pi^B_F= \varinjlim_{(a, A)\in B\downarrow F} \pi_A $

or in more explicit way (in terms of the domains categories):

$B\downarrow F= \varinjlim_{a: B\to F(A)} A\downarrow \mathcal{A}$. And then $(\pi^B_F)^{op}= \varinjlim_{(a, A)\in B\downarrow F} (\pi_A)^{op}$

**Then my question is:**

Let $\mathcal{C}=\varinjlim_{i\in I}\mathcal{C}_i$ a colimit of categories and $P: \mathcal{C}\to CAT$ a pseudofutor, the natural coproiections $\varepsilon_i: \mathcal{C}_i\to \mathcal{C}$ induce a morphism $e_i: \underrightarrow{LIM} (P\circ \varepsilon_i)\to \underrightarrow{LIM} P $, is this a pseudo-colimit? in other therm is EVER true that:

$\underrightarrow{LIM} P=\varinjlim_{i\in I}\ \underrightarrow{LIM} P_{|\mathcal{C}_i}$ ?

What for lax.colimit?

Bibliography: [G]: J , Giraud "Cohomologie non Abelienne"