This question came up when I was reading over some information about sheaves; specifically, that if $\mathscr{F}$ is a sheaf on the topological space $X$, $x\in X$, and $Z\subseteq X$, then $(\mathscr{F}|_Z )_x =\mathscr{F}_x$. I don't know if this is supposed to be trivial, and while it definitely seems to be a desirable property, I didn't see it as being obvious that it is true. After thinking for a while, I came to this conclusion:

Let $\mathfrak{I}$ be a directed category, and $F:\mathfrak{I}\rightarrow\mathfrak{C}$ a covariant functor; for $i\in \mathfrak{I}$, denote $F(i)$ by $A_i$. Fix a particular $I\in\mathfrak{I}$, and let $\mathfrak{J}$ be the full subcategory of $\mathfrak{I}$ satisfying $\mathrm{obj}(\mathfrak{J})=\lbrace j\in \mathfrak{I}\mid \mathrm{Mor}_\mathfrak{I}(I,j)\neq \emptyset\rbrace$. Then $\underrightarrow{\mathrm{lim}}_{i\in \mathfrak{I}}A_i =\underrightarrow{\mathrm{lim}}_{j\in \mathfrak{J}}A_j$.

I believe that this is true (well, I wrote a proof that convinced myself, anyway). If this is correct, it would explain why the statement about the stalks of the sheaves earlier is true. I was wondering if something more general can be said about when a colimit of a subcollection is equal to the colimit of the entire collection? Is there an even more general principle at work here, other than just a property of colimit?