  We generalize to dimension 2 the well-known fact that a colimit in a
1-dimensional slice is precisely the map from the colimit of the domains which
is induced by the universal property. We show two different approaches to this;
a more intuitive one, based on the reduction of the weighted 2-colimits to
oplax normal conical ones, and a more abstract one, based on an original
concept of colim-fibration and on an extension of the Grothendieck
construction. We find the need to consider lax slices, and prove results of
preservation, reflection and lifting of 2-colimits for the domain 2-functor
from a lax slice. The preservation result is shown by proving a general theorem
of $\F$-category theory, which states that a lax left adjoint preserves
appropriate colimits if the adjunction is strict on one side and is suitably
$\F$-categorical. Finally, we apply this theorem of preservation of 2-colimits
to the 2-functor of change of base along a split Grothendieck opfibration
between lax slices, after showing that it is such a left adjoint by laxifying
the proof that Conduch\'{e} functors are exponentiable. We conclude extending
the result of preservation of 2-colimits for the change of base 2-functor to
any finitely complete 2-category with a dense generator.
