Can the complement of a subset be realized as a limit or colimit? Let $X$ and $Y$ be two sets, and let $f:X\rightarrow Y$. Now consider the posets $(\mathcal{P}(X),\subseteq)$ and $(\mathcal{P}(Y),\subseteq)$ as categories. The induced functions $f^*$ (preimage function), $f_!$ (forward image function), and $f_*$ (the function which sends $B\subseteq X$ to the largest subset $C$ of $Y$ such that $f^*(C)\subseteq B$) can then be interpreted as functors between the said categories.
These functions also lead to a nice couple of adjunctions; namely, we have $f_!\dashv f^*\dashv f_*$
Since unions can be realized as colimits and intersections as limits, we get a nice categorical explanation for why $$f^*\left(\bigcup_{C\in\mathcal{C}}C\right)=\bigcup_{C\in\mathcal{C}}f^* (C)\quad\text{and}\quad f^*\left(\bigcap_{C\in\mathcal{C}}C\right)=\bigcap_{C\in\mathcal{C}}f^*(C)$$
However, we also have that $$f^*(Y-B)=X-f^*(B) \text{ for all } B\subseteq Y$$
This leads me to wonder the following:

Can complements be realized as a limit or a colimit?

Any help is appreciated.

As noted by Zhen, complements cannot be described via limits or colimits. Nevertheless, we have the following interpretation:
Recall the notions of monoidal categories and internal homs in monoidal categories.
In a monoidal poset, $\underline{\hom}(x,y)$ is the largest object such that $ \underline{\hom}(x,y) \cdot x \leq y$. If $y=0$ is an initial object (i.e. smallest element), then $\neg x := \underline{\hom}(x,0)$ is the largest object such that $(\neg x) \cdot x = 0$.
If $F : C \to D$ is a monoidal functor between closed monoidal posets, then there is a canonical morphism $F(\underline{\hom}(x,y)) \to \underline{\hom}(F(x),F(y))$ and $F$ is called closed when this is an isomorphism. If $F$ is closed and $F(0)=0$, then we get $F(\neg x)=\neg F(x)$.
If $F$ has a left adjoint $L$, then $F$ is closed if and only if the "projection formula" $L(x F(y))=L(x) y$ holds. This is easy to check and for some reason is called Frobenius reciprocity at the nlab.
The monoidal functor  $f^* : (P(Y),\subseteq,\cap) \to (P(X),\subseteq,\cap)$ is closed, because we have $f_!(x \cap f^*(y))=f_!(x) \cap y$.
