A morphism \varphi : \mathcal{F} \to \mathcal{G} of presheaves on X is a family of maps \varphi_U : \mathcal{F}(U) \to \mathcal{G}(U), one for each open set U, with \varphi_U(s)|_{V} = \varphi_V(s|_{V}) whenever V \subset U [1]. Since it commutes with restriction, \varphi sends germs to germs and gives a map \varphi_x : \mathcal{F}_x \to \mathcal{G}_x on every stalk. A morphism of sheaves is a morphism of the underlying presheaves.
A continuous map f : X \to Y turns sheaves on X into sheaves on Y: the push-forward f_{*} \mathcal{F} is the sheaf on Y with (f_{*} \mathcal{F})(V) = \mathcal{F}(f^{-1}(V)). For the inclusion i of a point p into X, the push-forward of \mathbb{R} is the skyscraper sheaf at p: (i_{*} \mathbb{R})(U) = \mathbb{R} when p \in U, and 0 when p \notin U. Taking the value at p is a morphism C \to i_{*} \mathbb{R} that sends f \in C(U) to f(p) when p \in U.
The kernel U \mapsto \ker \varphi_U of a morphism of sheaves is again a sheaf. A section whose restrictions to a cover all map to 0 maps to 0 by locality in \mathcal{G}, and compatible local sections of the kernel glue in \mathcal{F} to a section that maps to 0 for the same reason. The kernel of C \to i_{*} \mathbb{R} consists of the functions vanishing at p over open sets that contain p, and of all of C(U) over the others. In the figure the value appears exactly while the interval U contains p.
Images behave differently. Let X = \mathbb{C} \setminus \{0\} and let \mathcal{O}^{\times} be the sheaf of holomorphic functions without zeros, a sheaf of groups under multiplication. Squaring, g \mapsto g^2, is a morphism \mathcal{O}^{\times} \to \mathcal{O}^{\times}, and its image presheaf assigns to U the functions that are squares of functions in \mathcal{O}^{\times}(U) [1].
The function z is a square on every disk in X: such a disk misses a ray from 0, and on the plane cut along that ray z has a holomorphic square root. These local square roots cannot be chosen to agree on all the overlaps, since following a square root once around 0 changes its sign. So z is not a square on X, and the image presheaf does not satisfy the gluing condition.
The figure shows the real part of \sqrt{z} as a height over an annulus around 0. The two square roots form two sheets, which cross over the negative real axis. A point that goes once around 0 on the surface arrives on the other sheet, at -\sqrt{z}, and returns to its starting height only after a second turn.
The exponential behaves in the same way. The morphism \mathcal{O} \to \mathcal{O}^{\times}, g \mapsto e^{g}, reaches z on every disk in X, because z has a logarithm there, but not on X: going once around 0 adds 2\pi i to any logarithm of z [2].
Every presheaf \mathcal{F} has a sheafification \mathcal{F}^{+}. A section of \mathcal{F}^{+} over U is a choice of germs s_x \in \mathcal{F}_x, one for each x \in U, such that every point of U has a neighborhood W and a section t \in \mathcal{F}(W) with t_y = s_y for every y \in W [1]. Families of germs that come locally from sections satisfy both sheaf conditions.
There is a morphism \mathcal{F} \to \mathcal{F}^{+} that sends a section to its family of germs. Every morphism from \mathcal{F} to a sheaf factors through it in exactly one way, and the stalks do not change. The constant presheaf with value \mathbb{R} has the sheaf of locally constant functions as its sheafification, so over an open set with three connected components its sections form \mathbb{R}^3 instead of \mathbb{R}.
In the figure a section of the presheaf moves the three flat pieces together, while a section of the sheafification moves them separately. When a thin strip joins two of the components, the open set has two components, the pieces over the joined ones are forced to one height, and the sections form \mathbb{R}^2.
The image of a morphism of sheaves is the sheafification of its image presheaf, and the morphism is surjective when this image is all of \mathcal{G}, which happens exactly when every map \varphi_x on stalks is surjective [1]. Squaring on \mathcal{O}^{\times} is surjective in this sense, since every germ of a nonvanishing holomorphic function has a square root, while the map on sections over X misses z. Its kernel is the sheaf of locally constant functions with values 1 and -1.