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 $\mathcal{C}^{\mathrm{op}}\overset{\qquad c\quad\mapsto\quad \mathcal{D}(F(c),d)\qquad }{\longrightarrow}\mathsf{Set}$ is representable.

* Is there a usual technical term for a _representing object_ of this presheaf? Or rather, usual or not, have you ever _encountered_ a specialized term for this, and which do you _recommend_?

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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", I always used to call and denote it $\mathrm{law}(F)$, for "left-adjoint witness of $F$", since if a $\mathrm{law}(F)$ exists, then $F$ is a left-adjoint, and such an object is something of a certificate for its being a left-adjoint.
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* Do you agree that, strictly speaking, it would _not_ be in accordance with the definition of "the" in [the][1] to refer to "the law of $F$"? 


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After all, 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.
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  [1]: https://ncatlab.org/nlab/show/generalized+the