There is an orthogonal factorization system on the category Cat, whose left class is the class of bijective-on-objects functors, or “bo functors” and whose right class is the class of full and faithful functors, or “ff functors”.
This means that each functor decomposes as a composition of the form , where is bijective on objects and fully faithful; and if
is a commutative diagram with bijective on objects and essentially surjective, then there is a unique functor such that and .
In fact, this can be generalized to a square commuting up to invertible natural transformation, in which case one still concludes that but that , with the isomorphism composing with to give the original isomorphism. This means that this is an enhanced factorization system?.
This factorization system can be constructed using generalized kernels.