= Solution
For a <presheaf> $\mathcal F$, first form its <stalks> $\mathcal F_x=\varinjlim_{x\in U}\mathcal F(U)$. Define $\mathcal F^+(U)$ to consist of families $(s_x)_{x\in U}$, with $s_x\in\mathcal F_x$, which are locally represented by sections of $\mathcal F$: every $x\in U$ has a neighbourhood $V\subseteq U$ and $t\in\mathcal F(V)$ with $s_y=t_y$ for all $y\in V$. Addition is pointwise and restriction discards the components outside the smaller open set. The local representability condition is itself local, so compatible such families on an <open cover> glue uniquely. Thus $\mathcal F^+$ is a <sheaf of abelian groups>. This is <sheafification by locally representable germs>.
The canonical <presheaf morphism> is
$$
\boxed{\theta_U:\mathcal F(U)\longrightarrow\mathcal F^+(U),\qquad t\longmapsto(t_x)_{x\in U}.}
$$
The <universal property of sheafification> says that, for any <sheaf> $\mathcal H$, composition with $\theta$ gives a natural bijection
$$
\operatorname{Hom}(\mathcal F^+,\mathcal H)\cong\operatorname{Hom}_{\mathrm{pre}}(\mathcal F,\mathcal H).
$$
Indeed, locally representing a family by $t$ defines its image locally by the image of $t$ in $\mathcal H$. Equal <germs> give locally equal images, and the <sheaf gluing axiom> gives the unique global image. If $\mathcal F$ is already a <sheaf>, a section with all <germs> zero is zero, while every locally represented family glues to an actual section. Hence $\theta_U$ is both injective and surjective for every $U$, so \b[<sheafification> leaves a <sheaf> unchanged].
For a continuous map $f:X\to Y$, the <direct image sheaf> is
$$
\boxed{(f_*\mathcal F)(U)=\mathcal F(f^{-1}U).}
$$
An <open cover> pulls back to an <open cover>, so its <sheaf> axioms follow directly from those of $\mathcal F$. To construct the <inverse image sheaf>, first set
$$
P(V)=\varinjlim_{\substack{U\subseteq Y\text{ open}\\f(V)\subseteq U}}\mathcal G(U),\qquad f^{-1}\mathcal G=P^+.
$$
Restriction uses the inclusion of these neighbourhood systems when $V$ shrinks. The use of <sheafification> is important: $P$ need not itself be a <sheaf>. Continuity and the <stalk> construction give $(f^{-1}\mathcal G)_x\cong\mathcal G_{f(x)}$.
Given an <f-morphism of sheaves> $\phi$, define its map on <stalks> by
$$
\boxed{\phi_x:\mathcal G_{f(x)}\longrightarrow\mathcal F_x,\qquad[s,U]\longmapsto[\phi(U)(s),f^{-1}U].}
$$
If two representatives have the same <germ> at $f(x)$, they agree on a neighbourhood there. Restriction compatibility makes their images agree on its inverse image, a neighbourhood of $x$, proving well-definedness. The map is a group homomorphism. We index it by $x$, since different points over the same $f(x)$ have different target <stalks>.
For $s\in\mathcal G(U)$, its class in $P(f^{-1}U)$ and then in its <sheafification> defines the canonical <f-morphism of sheaves> $\theta(U)$. Its <germ> at $x$ is simply $s_{f(x)}$. To factor any $\phi$, take a section $t\in(f^{-1}\mathcal G)(V)$. Locally on an <open cover> $V=\bigcup V_a$, it comes from a section $s_a\in\mathcal G(U_a)$ with $V_a\subseteq f^{-1}U_a$. Define the prospective image on $V_a$ by
$$
\psi(t)|_{V_a}=\phi(U_a)(s_a)|_{V_a}.
$$
On an overlap, the representatives have the same inverse-image <germs>, so their images have the same <germs> by the maps $\phi_x$. Two <sheaf> sections with equal <germs> everywhere are equal. The local images therefore glue uniquely, independently of every representative and cover choice. This construction is additive and commutes with restrictions, giving a <sheaf morphism> $\psi:f^{-1}\mathcal G\to\mathcal F$ with
$$
\boxed{\phi(U)=\psi(f^{-1}U)\circ\theta(U).}
$$
Conversely any factorization must have the prescribed image on these locally generating sections, proving uniqueness. This is the <universal property of an inverse image sheaf>.
An <f-morphism of sheaves> is exactly a <sheaf morphism> $\mathcal G\to f_*\mathcal F$. Thus the two constructions give the <inverse-image direct-image adjunction>
$$
\boxed{\operatorname{Hom}_X(f^{-1}\mathcal G,\mathcal F)\cong\operatorname{Hom}_Y(\mathcal G,f_*\mathcal F).}
$$
Naturality follows from composing the local representatives and their images with morphisms in either <sheaf> variable. This is an adjunction for <sheaves> of abelian groups; it is not the tensor-adjusted pullback of <modules> on a ringed space.
Back to article page
