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First form the inverse image sheaf ${\varphi }^{-1}\mathcal{H}$ by sheafifying the presheafwhere $U$ runs through open subsets of $X$. It is naturally a module over ${\varphi }^{-1}{\mathcal{O}}_X$, not yet over ${\mathcal{O}}_Y$. The structure morphism ${\varphi }^{-1}{\mathcal{O}}_X\to {\mathcal{O}}_Y$ allows extension of scalars, giving the pullback of a sheaf of modules:The tensor product of sheaves includes its own sheafification. At $Q\in Y$ the construction becomesThis adjustment of the coefficient ring distinguishes the pullback from the underlying inverse image sheaf.

For a presheaf $\mathcal{F}$, first form its stalks ${\mathcal{F}}_x={\underrightarrow{\lim }}_{x\in U}\mathcal{F}(U)$. 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 isThe universal property of sheafification says that, for any sheaf $\mathcal{H}$, composition with $\theta$ gives a natural bijectionIndeed, 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 sheafification leaves a sheaf unchanged.

For a continuous map $f:X\to Y$, the direct image sheaf isAn 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 setRestriction 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 $\varphi$, define its map on stalks byIf 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 $\varphi$, 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 $V_a$ byOn an overlap, the representatives have the same inverse-image germs, so their images have the same germs by the maps ${\varphi }_x$. 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 $\psi :f^{-1}\mathcal{G}\to \mathcal{F}$ withConversely 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 $\mathcal{G}\to f_{\ast }\mathcal{F}$. Thus the two constructions give the inverse-image direct-image adjunctionNaturality 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.

First form the inverse image sheaf ${\varphi }^{-1}\mathcal{H}$ by applying sheafification toIt is a sheaf of modules over ${\varphi }^{-1}{\mathcal{O}}_Y$. The morphism of ringed spaces gives a ring map ${\varphi }^{-1}{\mathcal{O}}_Y\to {\mathcal{O}}_X$, so the pullback of a sheaf of modules isAt $x\in X$, writing $y=\varphi (x)$, its stalk is ${\mathcal{O}}_{X,x}{\otimes }_{{\mathcal{O}}_{Y,y}}{\mathcal{H}}_y$. The tensor construction makes the action of the structure sheaf explicit; taking the inverse image alone would not give the requested ${\mathcal{O}}_X$-module.

The tensor product of sheaves is the sheafification of the presheafwith restriction maps induced by those of $\mathcal{F}$, $\mathcal{G}$, and the structure sheaf of a scheme. Sending the germ of $f\otimes g$ to $f_P\otimes g_P$ gives a canonical mapIt is surjective because a finite sum of tensors of germs has representatives on one common neighborhood. It is injective because an equality in a tensor product of modules is witnessed by finitely many additive and balancing relations; all their germs, including the scalars in ${\mathcal{O}}_{X,P}$, can be represented and their equalities made valid on a common smaller neighborhood. Equivalently, a filtered colimit of modules commutes with these tensor products. Since sheafification preserves the stalk of a sheaf, this proves the asserted isomorphism.

For the pullback of a sheaf of modules, first form the inverse image sheaf ${\varphi }^{-1}\mathcal{F}$: sheafify $U\mapsto {\mathrm{colim}}_{V\supseteq \varphi (U)}\mathcal{F}(V)$. The morphism of sheaves of rings ${\varphi }^{-1}{\mathcal{O}}_Y\to {\mathcal{O}}_X$ then definesExactness of a sequence of sheaf of modules morphisms is checked on every stalk of a sheaf. Applying the right exactness of the tensor product of modules to the stalk sequence over ${\mathcal{O}}_{Y,\varphi (P)}$ gives the exact stalk sequence of the pullbacks. Thus ${\varphi }^{\ast }$ preserves sequences ending in zero, as required.

A quasi-coherent sheaf is a sheaf of modules locally of the form $\tilde{N}$ for a module $N$ over the coordinate ring of an affine neighborhood. On an affine variety $V$ with ring $B$, this associated sheaf is characterized on principal opens by $\tilde{N}(D(b))=N_b$, using localization of a module. Suppose $\mathcal{F}{|}_V\cong \tilde{N}$ and $U\subseteq {\varphi }^{-1}V$ is affine, with ring $C$. The induced algebra map $B\to C$ gives a canonical mapAt a point $P\in U$, both sides identify with ${\mathcal{O}}_{X,P}{\otimes }_{B}N$: on the right use ${\mathcal{F}}_{\varphi (P)}\cong {\mathcal{O}}_{Y,\varphi (P)}{\otimes }_{B}N$ and associativity of the tensor product of modules. The map is therefore an isomorphism on every stalk of a sheaf, hence an isomorphism of sheaves. Such $U$ cover $X$, proving that the pullback is a quasi-coherent sheaf.

Finally, on an affine open $V\subseteq Y$ with ring $B$, the presheaf defining $M_Y$ has, on $D(b)\subseteq V$, the valueConsequently $M_Y{|}_V\cong \widetilde{M{\otimes }_{R}B}$, so $M_Y$ is a quasi-coherent sheaf. The algebra map for $M_X$ is the composite $R\to \Gamma (Y,{\mathcal{O}}_Y)\to \Gamma (X,{\mathcal{O}}_X)$. Restriction of functions and the tensor-product construction give a canonical sheaf morphism ${\varphi }^{\ast }{M}_Y\to M_X$. At $P$, it is the associativity isomorphismIt is therefore a sheaf of modules isomorphism. In particular,

The tensor product of sheaves $\mathcal{F}{\otimes }_{{\mathcal{O}}_X}\mathcal{G}$ is the sheafification of the presheafEquivalently, its stalk at $x$ is ${\mathcal{F}}_x{\otimes }_{{\mathcal{O}}_{X,x}}{\mathcal{G}}_x$.

The direct image sheaf is defined on each open set $V\subseteq Y$ byThe pullback of a sheaf of modules iswhere $f^{-1}\mathcal{E}$ is the inverse image sheaf and the tensor product uses the structural morphism $f^{-1}{\mathcal{O}}_Y\to {\mathcal{O}}_X$ of the given ringed-space morphism.

The adjunction morphism $f^{-1}{f}_{\ast }\mathcal{F}\to \mathcal{F}$, together with $f^{-1}\mathcal{E}\to f^{\ast }\mathcal{E}$, givesAdjunction between the inverse image sheaf and direct image sheaf turns this into the projection formula for sheaves morphismOn local sections it sends a pure tensor $s\otimes e$ over $V\subseteq Y$ to $s\otimes f^{\#}e$ over $f^{-1}V$.

Whether this morphism is an isomorphism is local on $Y$. If $\mathcal{E}{|}_V\cong {\mathcal{O}}_V^{\oplus r}$ for finite $r$, its restriction becomes the canonical identificationThus the projection-formula morphism is an isomorphism whenever $\mathcal{E}$ is locally free of finite rank.

Every f-morphism of sheaves from $\mathcal{G}$ to $\mathcal{F}$ factors uniquely through the canonical f-morphism from $\mathcal{G}$ to its inverse image sheaf. A local inverse-image section is represented by sections of $\mathcal{G}$ on image neighbourhoods; their prescribed images glue in $\mathcal{F}$.