If actually has a left adjoint, then is a free -object on for every , and conversely if there exists a free -object on every then has a left adjoint. But individual free objects can exist without the whole left adjoint functor existing. In general, we have a “partially defined adjoint”, or -relative adjoint where is the inclusion of a full subcategory (on those objects admitting free objects).
More precisely: a free -object on consists of an object together with a morphism in such that for any other and morphism in , there exists a unique in with .
In other words, it is an initial object of the comma category . A free -object on is also sometimes called a universal arrow from to the functor . It can also be identified with a semi-final lift of an empty -structured sink.
Sometimes one says an object of is free (relative to a forgetful functor which is often tacitly understood) if there is some object of and some arrow that is initial in . For example, the Quillen-Suslin theorem says that finitely generated projective modules over polynomial algebras over a field are free; the tacit forgetful functor is from the category of modules over a polynomial algebra to . In this way, freeness is understood as a property of an object.
Similarly, a cofree object (or fascist object) is given by a cofree functor.
For more examples see at free construction.
A general way to construct free objects is with a transfinite construction of free algebras (in set-theoretic foundations), or with an inductive type or higher inductive type (in type-theoretic foundations).
free object, free resolution
flat object, flat resolution