The slice 2-category of a 2-category over an object is the 2-category whose objects are morphisms in , whose morphisms are triangles in that commute up to a specified isomorphism, and whose 2-cells are 2-cells in forming a commutative 2-diagram with the specified isomorphisms in triangles.
In particular, any 2-cell in must become an isomorphism in . This means that more information is lost when passing to slice 2-categories than for slice 1-categories, and slice 2-categories are not always well-behaved; they often fail to inherit useful properties of . Frequently a better replacement is the fibrational slice.
If has pullbacks, then for any there is a pullback functor . However, this does not make the assignation into a functor or , since there is no way to define it on 2-cells. This is one reason to use fibrational slices instead.
Just as in the 1-categorical case, the pullback functor always has a left adjoint given by composition with . However, cannot be expected to have a right adjoint for all maps , since this fails even in . It is true in when is a fibration or opfibration, however; see exponentials in a 2-category.