Yes, it's a general construction which is related to so-called Isbell conjugation.
Let $C$ be a small category. It is well-known that the free colimit cocompletion is given by the Yoneda embedding into presheaves on $C$, $y: C \to Set^{C^{op}}$. The presheaf category is also complete. Dually, the free limit-completion is given by the dual Yoneda embedding $y^{op}: C \to (Set^C)^{op}$. The co-presheaf category is also cocomplete.
Therefore there is a cocontinuous functor $L: Set^{C^{op}} \to (Set^C)^{op}$ which extends $y^{op}$ along $y$. This is a left adjoint; its right adjoint is the (unique up to isomorphism) functor $R: (Set^C)^{op} \to Set^{C^{op}}$ which extends $y$ continuously along $y^{op}$. This adjoint pair is called Isbell conjugation.
As is the case for any adjoint pair, this restricts to an adjoint equivalence between the full subcategories consisting, on one side, of objects $F$ of $Set^{C^{op}}$ such that the unit component $F \to R L F$ is an iso, and on the other side of objects $G$ of $(Set^C)^{op}$ such that the counit $L R G \to G$ is an iso. Either side of this equivalence gives the Dedekind-MacNeille completion of $C$. By the Yoneda lemma, $y: C \to Set^{C^{op}}$ factors through the full subcategory of DM objects as a functor $C \to DM(C)$ which preserves any limits that exist in $C$, and dually $y^{op}: C \to (Set^C)^{op}$ factors as the same functor $C \to DM(C)$ which preserves any colimits that exist in $C$.
Edit: Perhaps it might help to spell this out a little more. The classical Dedekind-MacNeille completion is obtained by taking fixed points of a Galois connection between upward-closed sets and downward-closed sets of a poset $P$. So, if $A$ is downward-closed (i.e., a functor $A: P^{op} \to \mathbf{2}$), and $B: P \to \mathbf{2}$ is upward-closed, we define
$$A^u = \{p \in P: \forall_{x \in P} x \in A \Rightarrow x \leq p\}$$
$$B^d = \{q \in P: \forall_{y \in P} y \in B \Rightarrow q \leq y\}$$
and one has
$$A \subseteq B^d \qquad \text{iff} \qquad A \times B \subseteq (\leq) \qquad \text{iff} \qquad B \subseteq A^u$$
We thus have an adjunction
$$(L = (-)^u: \mathbf{2}^{P^{op}} \to (\mathbf{2}^P)^{op}) \qquad \dashv \qquad (R = (-)^d: (\mathbf{2}^P)^{op} \to \mathbf{2}^{P^{op}})$$
and the poset of downward-closed sets $A$ for which $A = (A^u)^d$ is isomorphic to the poset of upward-closed sets $B$ for which $(B^d)^u = B$.
All of this can be "categorified" so as to hold in a general enriched setting, where the base of enrichment is a complete, cocomplete, symmetric monoidal closed category $V$. We may take for example $V = Set$. Analogous to the formation of $B^d$, we may define half of the Isbell conjugation $R: (Set^C)^{op} \to Set^{C^{op}}$ by the formula
$$R(G) = \int_{d \in C} \hom(-, d)^{G(d)}$$
where $\hom$ plays the role of the poset relation $\leq$, exponentiation or cotensor plays the role of the implication operator, and the end plays the role of the universal quantifier. The other half $L: Set^{C^{op}} \to (Set^C)^{op}$ is also defined, at the object level, by
$$L(F) = \int_{c \in C} \hom(c, -)^{F(c)}$$
(the right-hand side is a set-valued functor $C \to Set$; when we interpret this in $(Set^C)^{op}$, the end is interpreted as a coend, and the cotensor is interpreted as a tensor). In any event, given $F: C^{op} \to Set$ and $G: C \to Set$, we have natural bijections between morphisms
$$\{F \to R(G)\} \qquad \cong \qquad \{F \times G \to \hom\} \qquad \cong \qquad \{G \to L(F)\}$$
and the analogue of the MacNeille completion is obtained by taking "fixed points" of the adjunction $L \dashv R$, as described above by full subcategories where the unit and counit $F \to RLF$ and $LRG \to G$ become isomorphisms. These full subcategories are equivalent; one side of the equivalence is complete because it is the category of algebras for an idempotent monad associated with $RL$, and the other side is cocomplete because it is the category of coalgebras for an idempotent comonad associated with $LR$, and thus both sides are complete and cocomplete.
Edit: It has been pointed out that there is a mistake in the argument at the end of the prior edit, asserting that the fixed points of the monad coincide with the algebras of an associated idempotent monad. See Michal's answer (posted 9/13/2013).
As Denis-Charles says in the comments, the best way to handle this is to replace the presheaf category $\mathbf{Set}^{C^{\mathrm{op}}}$ by the full subcategory $\hat{C}$ of small presheaves. By definition, a presheaf is small if it satisfies any of these equivalent conditions:
it is a small colimit of representables;
it is the left Kan extension of its restriction to some small full subcategory of $C$;
it is the left Kan extension of some presheaf on some small category along some functor into $C$.
Every presheaf on a small category is small. But, for instance, a presheaf on a large discrete category is small iff its support is small; hence the terminal presheaf is not small.
The functor $C \mapsto \hat{C}$ is left adjoint to the forgetful functor
$$
(\text{cocomplete locally small categories}) \to (\text{locally small categories}),
$$
in a suitable 2-categorical sense.
A standard reference for this is:
Brian J. Day and Stephen Lack. Limits of small functors. Journal of Pure and Applied Algebra 210 (2007), 651–663.
But it goes back further than 2007. The introduction to Day and Lack's paper recounts some of the history.
Best Answer
The question is: Given a functor $F : A^{op} \to \mathsf{Set}$, how do we call an object $?(F)$ in $A$ satisfying the universal property
$\hom(?(F),X) \cong \hom(F,\hom(-,X))$
for all $X \in A$? Some people call it a corepresenting object of $F$. The reason is that a representing object of $F$ is some object $!(F)$ satisfying $\hom(X,!(F)) \cong \hom(\hom(-,X),F)$, since the left hand side simplifies to $F(X)$ by the Yoneda Lemma. Remark that every representing object is also a corepresenting object.
If $F$ is a moduli problem in algebraic geometry, then $?(F)$ with some additional assumptions is usually also called a coarse moduli space (whereas $!(F)$ is the fine moduli space). One of the many references is Definition 2.1. (2) in Adrian Langer's "Moduli Spaces Of Sheaves On Higher Dimensional Varieties", as well as Definition 2.2.1 in "The Geometry of Moduli Spaces of Sheaves" by Huybrecht and Lehn. Perhaps someone can add the original reference.