Let $F$ and $G$ be sheaves of sets on some topological space $X$, and let $x\in X$. People often say that each natural transformation $F\to G$ induces a map on the stalks $F_x\to G_x$, but they don't define precisely what that map is. Apparently it's trivial for most people, but since I am stupid I have to think about that.
Recall that the stalk of $F$ at $x$ is the colimit of all $F(U)$ where $U\ni x$.
Also, recall that by definition the colimit construction $C^J\to C$ is left adjoint to the diagonal $C\to C^J$. In particular it is itself a functor: if $F', G'\colon J\to C$, then each natural transformation $F'\to G'$ induces a map from the colimit of $F'$ to the colimit of $G'$. (*)
Now let $F\to G$ be a natural transformation between the sheaves $F$ and $G$. This induces a natural transformation between $F', G'\colon J\to C$, where $J:=\{U\in\mathrm{Open}(X)\mid U\ni x\}$, $C:=\mathrm{Set}$, and $F'$ and $G'$ are the restrictions of $F$ and $G$ to $J$. This in turn induces by (*) a map from the colimit of $F'$ to the colimit of $G'$. But the colimit of $F'$ is by definition the stalk of $F$ at $x$, similarly for $G$. Hence we get a map $F_x\to G_x$.
Is the map I just described equal to the map people call the "canonical map" $F_x\to G_x$ induced by the natural transformation $F\to G$?
Just to be a little more concrete, an element of $F_p$ is a pair $(U, f)$ where $f \in F(U)$ and where we identify two such pairs $(f, U)$ and $(g, V)$ if $f = g$ on some subset of $U \cap V$ that still contains $p$.
Given such a pair $(U, f)$ we get a section $\phi(f)$ of $G(U)$ since we have a sheaf map $\phi: F \to G$. Check that the pair $(V, \phi(f))$ is well-defined as an element of $G_p$.