TITLE: Direct and inverse limits of sheaves
QUESTION [5 upvotes]: Is the direct limit of sheaves a sheaf? Is the inverse limit of sheaves a sheaf? I guess another way of saying it is whether sheafification commutes with direct limit or inverse limit. While we are at it, does sheafification commutes with direct sum or product?

REPLY [4 votes]: Keenan Kidwell has answered your main question but I think I can add a little exposition earned from doing all the $\textrm{Ch II}\S1$ exercises in Hartshorne.
A finite (presheaf) product or coproduct of sheaves is again a sheaf, since $\mathscr{F} \oplus_{\mathsf{PSh}} \mathscr{G} \cong \mathscr{F} \times_{\mathsf{PSh}} \mathscr{G}$. Note that in general even finite colimits of sheaves have to be taken in $\mathsf{PSh}$ and then sheafified, e.g. cokernels.
However even a countably infinite family of sheaves can have the property that the direct sum(as presheaves) is not a sheaf! Take our space $\mathbb{N}_{\geq 0}$ with the discrete topology, and let $\mathscr{F}_i$ for $i \geq 0$ be the skyscraper sheaf with stalk $\mathscr{F}_{i,i}=\mathbb{Z}$ and zero elsewhere. Now we see that $\oplus_{i,\mathsf{PSh}} \mathscr{F}_i$ isn't a sheaf! Take the open sets $\{j\}_{j \geq 0}$ and the compatible family of sections $i_j(j)$ for $j \in \mathscr{F}_j(\{j\}) \cong \mathbb{Z}$ and where $i_j : \mathscr{F}_j \hookrightarrow \oplus_{i,\mathsf{PSh}} \mathscr{F}_{i}$.
EDIT: It occurs to me here that it's easier to show in general that if $\{ U_i \}$ is a family of disjoint open sets then $\mathscr{F}(\cup U_i) = \prod_i \mathscr{F}(U_i)$ if $\mathscr{F}$ is a sheaf. This immediately shows that this direct sum is not a sheaf.
It's a little dense, but if you do your part to break up that paragraph you'll see this family has no gluing, precisely because direct sums only have finitely many nonzero terms.