In a concrete category with images one can choose a representative of a subfunctor where the components of are genuine inclusions of the underlying sets; then a subfunctor is just a natural transformation whose components are inclusions. The naturality in terms of concrete inclusions just says that for all , . If the set-theoretic circumstances allow consideration of a category of functors, then a subfunctor is a subobject in such a category.
A subfunctor of the identity in a category with images is an often used case: it amounts to a natural assignment of a subobject to each object in . For concrete categories with images then .
Revised on March 29, 2011 23:35:27
by Toby Bartels