A functor is called **bijective on objects**, or **bo**, if it is a bijection on objects.

One reason bo functors are important is because together with full and faithful functors they form an orthogonal factorization system on Cat; see bo-ff factorization system. This factorization system can also be constructed using a generalized kernel.

To be more in accord with the principle of equivalence, one could require that the functor be bijective on objects only up to isomorphism; that is, it is essentially surjective and full on isomorphisms. However, from the point of view of factorization systems, the version of the concept of a bo functor which is in accord with the principle of equivalence is nothing more or less than an essentially surjective functor, since essentially surjective functors and ff functors form a bicategorical factorization system on the bicategory $Cat$.

R. Street in *Categorical and combinatorial aspects of descent theory* proves

Proposition. A functor is bijective on objects if and only if it exhibits its codomain as the (2-categorical) codescent object of some simplicial category.

This can be generalized to any regular 2-category.

**basic properties of…**

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