Technical term for representing object of a presheaf determined by a left-adjoint?

Question

If $\mathcal{D}$ is a locally-small category, then a functor $F\colon\mathcal{C}\rightarrow\mathcal{D}$ has a right-adjoint if and only if for each object $d$ of $D$, the presheaf $$\mathcal{C}^{\mathrm{op}}\xrightarrow[c\,\mapsto\, \mathcal{D}(F(c),d) ]{}\mathsf{Set}$$ is representable.

• Is there a usual technical term for, given $d$, a representing object of the aforementioned presheaf? Or rather, usual or not, have you ever encountered a specialized term for this, and which do you recommend?

_{Remarks. Whether any specialized term is needed, is debatable of course, and not the question. There are situations where it is useful to have a term for it, to facilitate discussing the issue, in particular in expositions. Saying ``representing object of $F$'' would be nonsensical. Of course, one can just describe it the way I did, but is there a specialized technical term for this representing object?
For want of a standard term, and despite the connotations of "law" and similarity to another, usual notation, I always used to call and denote "the" family}

$\mathrm{Ob}(\mathcal{D})\ni d\mapsto $(a representing object of the relevant presheaf) $\in$ $\mathrm{Ob}(\mathcal{C})$

"the" $\mathrm{Law}(F)$, for "left-adjoint witnesses of $F$", since if a family $\mathrm{Law}(F)$ exists, then $F$ is a left-adjoint, and such a family of objects is something of a certificate for its being a left-adjoint.

• Do you agree that, strictly speaking, it would not be in accordance with the definition of "the" in the to refer to "the law of $F$"?

_{After all, each "component" of $\mathrm{Law}(F)$ is determined only up to isomorphism in the category $\mathcal{D}$, and the feature of any two laws being determined up to unique isomorphism seems to be totally lacking.}

Answer 1

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.

Answer 2

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.