As requested, I'll turn my comments into an answer. There were two questions, the first being what we call the representing object if a presheaf $c \mapsto \mathcal{D}(F c, d)$ is representable, and the second being whether the "the" in "the representing object" is justified.

Based on some other recent discussions, it's the second question that I thought more critical to address. My answer is that a representation consists not only of a representing object $c'$ but also a specified isomorphism $\mathcal{C}(-, c') \stackrel{\sim}{\to} \mathcal{D}(F-, d)$ (as opposed to an object $c'$ with the property that there exists such an isomorphism). By the Yoneda lemma, the specified isomorphism is given by an element $\theta \in \mathcal{D}(Fc', d)$, variously called a universal element or a universal arrow $\theta: Fc' \to d$. Thus Mac Lane in Categories for the Working Mathematician speaks of a universal arrow from $F$ to $d$ as consisting of a certain ordered pair $(c' \in Ob(\mathcal{C}), \theta: F c' \to d)$, one that exhibits or witnesses the representability of $\mathcal{D}(F-, d)$, or equivalently, as "the" terminal object of a comma category $F \downarrow d$.

(That comma category formulation is technically useful, as not only does it justify the word "the" in that nLab sense -- terminal objects being always unique up to unique isomorphism -- but re-orients attention to the fact that all universal mapping problems are about is constructing initial or terminal objects in suitable categories. So if you read for example about adjoint functor theorems, that's how you should interpret what is going on in those technical hypotheses like solution-set conditions: one is trying to replace the construction of an initial object in some (usually large) category $\mathcal{A}$ as the limit of the large diagram $1_\mathcal{A}: \mathcal{A} \to \mathcal{A}$ by a limit over a small diagram that happens to be cofinal in the large diagram. That's all it is.)

Getting back to terminology and what you call the representing object: you could either follow Mac Lane and refer instead to the universal arrow from $F$ to $d$, which is fairly short and clear and has the right emphasis, or you could adapt the suggestion made by Mike Shulman and say something like "the cofree object (relative to $F$) cogenerated by $d$". As usual in such things, that's a slight abuse of language since, analogously, we don't consider a free group on a set $X$ as a just a group $G$, but rather as a group $G$ together with a function $X \to UG$ satisfying the appropriate universal property. But as long as that language abuse is clearly understood, that alternative terminology seems acceptable.

As requested, I'll turn my comments into an answer. There were two questions, the first being what we call the representing object if a presheaf $c \mapsto \mathcal{D}(F c, d)$ is representable, and the second being whether the "the" in "the representing object" is justified.

Based on some other recent discussions, it's the second question that I thought more critical to address. My answer is that a representation consists not only of a representing object $c'$ but also a specified isomorphism $\mathcal{C}(-, c') \stackrel{\sim}{\to} \mathcal{D}(F-, d)$ (as opposed to an object $c'$ with the property that there exists such an isomorphism). By the Yoneda lemma, the specified isomorphism is given by an element $\theta \in \mathcal{D}(Fc', d)$, variously called a universal element or a universal arrow $\theta: Fc' \to d$. Thus Mac Lane in Categories for the Working Mathematician speaks of a universal arrow from $F$ to $d$ as consisting of a certain ordered pair $(c' \in Ob(\mathcal{C}), \theta: F c' \to d)$, one that exhibits or witnesses the representability of $\mathcal{D}(F-, d)$, or equivalently, as "the" terminal object of a comma category $F \downarrow d$.

(That comma category formulation is technically useful, as not only does it justify the word "the" in that nLab sense -- terminal objects being always unique up to unique isomorphism -- but re-orients attention to the fact that all universal mapping problems are about is constructing initial or terminal objects in suitable categories. So if you read for example about adjoint functor theorems, that's how you should interpret what is going on in those technical hypotheses like solution-set conditions: one is trying to replace the construction of an initial object in some (usually large) category $\mathcal{A}$ as the limit of the large diagram $1_\mathcal{A}: \mathcal{A} \to \mathcal{A}$ by a limit over a small diagram that happens to be cofinal in the large diagram. That's all it is.)

Getting back to terminology and what you call the representing object: you could either follow Mac Lane and refer instead to the universal arrow from $F$ to $d$, which is fairly short and clear and has the right emphasis, or you could adapt the suggestion made by Mike Shulman and say something like "the cofree object (relative to $F$) cogenerated by $d$" (or "on $d$" or "over $d$", leaving off the "cogenerated"). As usual in such things, that's a slight abuse of language since, analogously, we don't consider a free group on a set $X$ as a just a group $G$, but rather as a group $G$ together with a function $X \to UG$ satisfying the appropriate universal property. But as long as that language abuse is clearly understood, that alternative terminology seems acceptable.