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.

Similarly, a cofree object (or fascist object) is given by a cofree functor.