Canonical example of a cosheaf

So you are probably requiring a more general sort of cosheaf -- I don't really know what you mean by "running the sheaf functor twice" -- but let me maybe provide a simpler description of cosheaves that might answer your question.

One definition of a cosheaf is simply a functor

$F:\mathrm{Open}(X)\to\mathcal{C}$

that sends colimits (unions) to colimits. By formality, we can say this is a sheaf valued in the opposite category

$F:\mathrm{Open}(X)^{op}\to\mathcal{C}^{op}.$

There are a couple of "canonical" examples of cosheaves. One is the cosheaf of compactly supported real valued functions on a space:

$U\rightsquigarrow \{f:U\to\mathbb{R}|\mathrm{supp}(f) \, \mathrm{cpt}\}$

Where the extension function is to extend by zero and partitions of unity should force the cosheaf axiom to hold.

More abstractly, using the colimit-preserving definition, any continuous map of spaces:

$f:X\to Y$

we can build a cosheaf of spaces on $Y$, by assigning to each open set $U\subset Y$

$U\rightsquigarrow f^{-1}(U) \qquad U\cup V \rightsquigarrow f^{-1}(U)\cup f^{-1}(V).$

Another, very closely related canonical example of a cosheaf, is to take $\pi_0(f^{-1}(U))$ and as long as $Y$ is locally connected, this will be a cosheaf. See Jon Woolf's paper The Fundamental Category of a Stratified Space and appendix B in there.

Finally, due to an observation of Bob MacPherson, if we think of a cell complex $X$ as a category with objects the cells $\sigma\in X$ with morphisms given by the face relation $\sigma\subseteq \bar{\tau}$, then a constructible sheaf is equivalent to a functor $F:X\to\mathrm{Vect}$ and a constructible cosheaf $F:X^{op}\to\mathrm{Vect}$. These gadgets are bona fide sheaves and cosheaves in the Alexandrov topology on the associated face relation poset, i.e. open sets are $U\subset X$ such that $x\in U$ $x\leq y$ implies $y\in U$.

Finally, I should say that some of my thesis work applies cosheaves to Morse theory, persistent homology and sensor networks, which should provide some more intuitive examples.