  We focus on two factorization systems for opfibrations in the 2-category
Fib(B) of fibrations over a fixed base category B. The first one is the
internal version of the so called comprehensive factorization, where the right
orthogonal class is given by internal discrete opfibrations. The second one has
as its right orthogonal class internal opfibrations in groupoids, i.e. with
groupoidal fibres. These factorizations can be obtained by means of a single
step 2-colimit. Namely, their left orthogonal parts are nothing but suitable
coidentifiers and coinverters respectively. We will show how these results
follow from their analogues in Cat. To this end, we first provide suitable
conditions on a 2-category C, allowing the transfer of the construction of
coinverters and coidentifiers from C to Fib(B).

Category Theory