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 <b>Isbell conjugation</b>.

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 $C$.

<b>Edit:</b> 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

$$A \subseteq B^d \qquad \text{iff} \qquad A \times B \subseteq (\leq) \qquad \text{iff} \qquad B \subseteq A^u$$

$$(L = (-)^u: \mathbf{2}^{P^{op}} \to (\mathbf{2}^P)^{op}) \qquad \dashv \qquad (R = (-)^d: (\mathbf{2}^P)^{op} \to \mathbf{2}^{P^{op}})$$

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.