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Tensoring sections on $f^{-1}V$ gives a balanced pairing of the two direct image sheaves on $V$. The universal property of the tensor product of modules and sheafification produce the displayed morphism. Composing with the pullback-direct-image adjunction unit gives the projection formula for sheaves. The comparison itself need not be an isomorphism.

Put $D=\mathbb{Q}/\mathbb{Z}$. The character module $M^{\ast }={\mathrm{Hom}}_{\mathbb{Z}}(M,D)$ has the natural actionAdditivity is immediate, and $a(b\varphi )(m)=\varphi (bam)=\varphi (abm)$, using commutativity of $A$. The identity of $A$ acts as the identity. We establish the extension and duality tools used in the four implications, so no flatness or injectivity criterion is assumed without proof.

An injective module $E$ is one for which every homomorphism from a submodule $U\subseteq V$ extends to $V$. The Baer criterion says it suffices to test the inclusions of ideals $I\subseteq A$. Necessity follows directly from the definition. For sufficiency, suppose every $I\to E$ extends to $A$. Given $h:U\to E$, use Zorn's lemma to choose a maximal extension $(W,h_W)$ inside $V$; extensions along a chain have a well-defined union. If $v\in V\setminus W$, letThis is an ideal and an $A$-linear map. Its extension to $A$ has the form $a\mapsto ae$ for some $e\in E$. Thenis well-defined: if two expressions represent the same element, their coefficient difference lies in $I$, where the two prescribed values agree. It extends $h_W$ to the strictly larger module $W+Av$, a contradiction. Thus $W=V$, proving the Baer criterion for arbitrary modules and ideals.

The group $D$ is divisible. A homomorphism $n\mathbb{Z}\to D$, for $n>0$, extends to $\mathbb{Z}$ by choosing $d\in D$ with $nd$ equal to its value at $n$; the zero ideal causes no restriction. The proved Baer criterion over $\mathbb{Z}$ therefore makes $D$ injective. It is also a cogenerator: a nonzero element $x$ of an abelian group generates either a finite cyclic group of order $n>1$, which maps $x$ to $1/n+\mathbb{Z}$, or an infinite cyclic group, which maps $x$ to $1/2+\mathbb{Z}$. Injectivity extends this character to the whole group. Thus every nonzero element is detected by an additive character.

It follows that a group homomorphism $u:T\to U$ is injective exactly whenis surjective. For an injection, extension into the injective group $D$ proves surjectivity. Conversely, a character nonzero on an element of $\ker u$ cannot be the pullback of a character on $U$. This is the detection property of the injective cogenerator of abelian groups.

Finally, for any $A$-module $N$ there is a natural character form of the Tensor-hom adjunction:An $A$-linear map $h$ gives the pairing $(n,m)\mapsto h(n)(m)$. It is balanced because $h(an)(m)=(ah(n))(m)=h(n)(am)$. Conversely a balanced additive pairing determines $h$ by this formula. The universal property of the tensor product of modules makes these constructions mutually inverse, and compatible with restriction maps.

The opening change-of-rings tensor quotient sends $m{\otimes }_{R}n$ to $m{\otimes }_{A}n$. The target pairing is bilinear and $R$-balanced because $\theta (r)m{\otimes }_{A}n=m{\otimes }_A\theta (r)n$. On the source, let $A$ act through the first factor; commutativity makes that action compatible with the $R$-balancing relations, and the map is $A$-linear. It is surjective: the target is the quotient imposing the additional relations $am{\otimes }_{R}n=m{\otimes }_{R}an$ for every $a\in A$. The construction commutes with module homomorphisms in both variables.

The pullback-direct-image adjunction unit is obtained locally by pulling a section $h\in \mathcal{H}(V)$ to ${\varphi }^{-1}V$ and sending it to $1\otimes h$ in ${\varphi }^{\ast }\mathcal{H}({\varphi }^{-1}V)$. These maps respect restrictions and the ${\mathcal{O}}_Y$-actions, hence defineFor $\mathcal{H}={\mathcal{O}}_Y$, multiplication identifies ${\varphi }^{\ast }{\mathcal{O}}_Y$ with ${\mathcal{O}}_X$, and ${\eta }_{{\mathcal{O}}_Y}$ is precisely the structure morphism ${\varphi }^{\#}:{\mathcal{O}}_Y\to {\varphi }_{\ast }{\mathcal{O}}_X$.

To construct the direct-image tensor comparison, on $V\subseteq Y$ send $s\otimes t$, with $s\in \mathcal{F}({\varphi }^{-1}V)$ and $t\in \mathcal{G}({\varphi }^{-1}V)$, to its tensor section of $\mathcal{F}{\otimes }_{{\mathcal{O}}_X}\mathcal{G}$. This pairing is ${\mathcal{O}}_Y(V)$-balanced through ${\varphi }^{\#}$ and is compatible with restrictions. The universal property of the tensor product of modules and sheafification therefore giveApply this with $\mathcal{G}={\varphi }^{\ast }\mathcal{H}$ after tensoring the unit ${\eta }_{\mathcal{H}}$ with ${\varphi }_{\ast }\mathcal{F}$. The composite is the map in the projection formula for sheaves. If $\mathcal{H}{|}_V\cong {\mathcal{O}}_V^r$, then ${\varphi }^{\ast }\mathcal{H}{|}_{{\varphi }^{-1}V}\cong {\mathcal{O}}_{{\varphi }^{-1}V}^r$, and the composite identifies withthe identity on the $r$ components. Thus it is an isomorphism for a locally free sheaf of finite rank. Neither $\mathcal{F}$ nor $\mathcal{G}$ was assumed quasi-coherent.

For $R$-modules $M,N$, the tensor product of modules is an $R$-module $M{\otimes }_{R}N$ together with the balanced mapsuch that every balanced map $b:M\times N\to P$ factors through one unique $R$-linear map $\tilde{b}$:Equivalently,naturally in $P$. This is the universal property of the tensor product of modules.

The Tensor-hom adjunction is the natural isomorphismFor an R-module homomorphism $f$, it is given explicitly byConversely, an $R$-linear map $g:M\to {\mathrm{Hom}}_R(N,P)$ determines the balanced map $(m,n)\mapsto g(m)(n)$, so the universal property of the tensor product of modules givesThese formulas are inverse to each other because pure tensors generate the tensor product of modules.

A universal property characterizes an object by a prescribed family of morphisms and a unique factorization of every competing family. It determines that object up to a unique compatible isomorphism. Examples include the pushout in a category and the universal property of the tensor product of modules.