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We study the 2-category of elements from an abstract point of view. Wegeneralize to dimension 2 the well-known result that the category of elementscan be captured by a comma object that also exhibits a pointwise left Kanextension. For this, we propose an original definition of pointwise Kanextension along a discrete 2-opfibration in the lax 3-category of 2-categories,2-functors, lax natural transformations and modifications. Such definition usescartesian-marked lax limits, which are an alternative to weighted 2-limits. Weshow that a pointwise Kan extension along a discrete 2-opfibration is always aweak one as well. The proof is based on an original generalization of theparametrized Yoneda lemma which is as lax as it can be.

Pointwise Kan extensions along 2-fibrations and the 2-category of elements