Todd Trimble’s answer makes very good points about representing objects/arrows in general. However, in your specific case — an object $c \in \newcommand{\D}{\mathbf{D}}\C$ such that $\newcommand{\C}{\mathbf{C}}\C(x,c) \cong \D(Fx,d)$, naturally in $x \in \C$ — it is generally known as the cofree object on $d$, as Mike Shulman mentions in comments.

This terminology is absolutely standard in the case where $F$ is some kind of forgetful functor; one has for instance cofree comodules over various sorts of Hopf-/bi-/co-algebras (in print, see e.g. Prop. 2.5.8 of Hovey’s book Model Categories), cofree coalgebras, and cofree comonads. For arbitrary $F$, it’s slightly less universal, but I’ve certainly heard it — and if one wants some term for this purpose, it seems like the obvious choice.

Todd Trimble’s answer makes very good points about representing objects/arrows in general. However, in your specific case — an object $c \in \newcommand{\C}{\mathbf{C}}\newcommand{\D}{\mathbf{D}}\C$ such that $\C(x,c) \cong \D(Fx,d)$, naturally in $x \in \C$ — it is generally known as the cofree object on $d$, as Mike Shulman mentions in comments.

This terminology is absolutely standard in the case where $F$ is some kind of forgetful functor; one has for instance cofree comodules over various sorts of Hopf-/bi-/co-algebras (in print, see e.g. Prop. 2.5.8 of Hovey’s book Model Categories), cofree coalgebras, and cofree comonads. For arbitrary $F$, it’s slightly less universal, but I’ve certainly heard it — and if one wants some term for this purpose, it seems like the obvious choice.