If $U$ actually has a left adjoint, then $F(x)$ is a free $C$-object on $x$ for every $x$, and conversely if there exists a free $C$-object on every $x\in D$ then $U$ 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 $J$-relative adjoint where $J$ is the inclusion of a full subcategory (on those objects admitting free objects).

More precisely: a free $C$-object on $x$ consists of an object $y\in C$ together with a morphism$\eta_x \colon x\to U y$ in $D$ such that for any other $z\in C$ and morphism $f\colon x\to U z$ in $D$, there exists a unique $g\colon y\to z$ in $C$ with $U(g) \circ \eta_x = f$.

In other words, it is an initial object of the comma category$(x/U)$. A free $C$-object on $x$ is also sometimes called a universal arrow from $x$ to the functor $U$. It can also be identified with a semi-final lift of an empty $U$-structured sink.

Sometimes one says an object $c$ of $C$ is free (relative to a forgetful functor $U: C \to D$ which is often tacitly understood) if there is some object $x$ of $D$ and some arrow $x \to U c$ that is initial in $(x/U)$. For example, the Quillen-Suslin theorem says that finitely generatedprojective modules over polynomial algebras over a field are free; the tacit forgetful functor is from the category of modules over a polynomial algebra to $Set$. In this way, freeness is understood as a property of an object.

Similarly, a cofree object is given by a cofree functor.