Corestriction

In mathematics, a corestriction of a function is a notion analogous to the notion of a restriction of a function. The duality prefix co- here denotes that while the restriction changes the domain to a subset, the corestriction changes the codomain to a subset. However, the notions are not categorically dual.
Given any subset formula_1 we can consider the corresponding inclusion of sets formula_2 as a function. Then for any function formula_3, the restriction formula_4 of a function formula_5 onto formula_6 can be defined as the composition formula_7. 
Analogously, for an inclusion formula_8 the corestriction formula_9 of formula_5 onto formula_6 is the unique
function formula_12 such that there is a decomposition formula_13. The corestriction exists if and only if formula_14 contains the image of formula_5. In particular, the corestriction onto the image always exists and it is sometimes simply called the corestriction of formula_5. More generally, one can consider corestriction of a morphism in general categories with images. The term is well known in category theory, while rarely used in print.
Andreottiintroduces the above notion under the name coastriction, while the name corestriction reserves to the notion categorically dual to the notion of a restriction. Namely, if formula_17 is a surjection of sets (that is a quotient map) then Andreotti considers the composition formula_18, which surely always exists.