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We generalize to dimension 2 the well-known fact that a colimit in a1-dimensional slice is precisely the map from the colimit of the domains whichis 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 tooplax normal conical ones, and a more abstract one, based on an originalconcept of colim-fibration and on an extension of the Grothendieckconstruction. We find the need to consider lax slices, and prove results ofpreservation, reflection and lifting of 2-colimits for the domain 2-functorfrom a lax slice. The preservation result is shown by proving a general theoremof $\F$-category theory, which states that a lax left adjoint preservesappropriate colimits if the adjunction is strict on one side and is suitably$\F$-categorical. Finally, we apply this theorem of preservation of 2-colimitsto the 2-functor of change of base along a split Grothendieck opfibrationbetween lax slices, after showing that it is such a left adjoint by laxifyingthe proof that Conduch\'{e} functors are exponentiable. We conclude extendingthe result of preservation of 2-colimits for the change of base 2-functor toany finitely complete 2-category with a dense generator.

Colimits in 2-dimensional slices