Addition of Cartier divisors is tensor product of their sheaves

Sum and inverse equations. On the common cover, $D+E$ is represented by the equations $f_i{g}_i$ and $-D$ by $f_i^{-1}$, and the associated sheaves satisfy ${\mathcal{O}}_X(D+E){\mathrm{\mid }}_{U_i}=(f_i{g}_i{)}^{-1}{\mathcal{O}}_{U_i}$, ${\mathcal{O}}_X(D){\mathrm{\mid }}_{U_i}=f_i^{-1}{\mathcal{O}}_{U_i}$, ${\mathcal{O}}_X(E){\mathrm{\mid }}_{U_i}=g_i^{-1}{\mathcal{O}}_{U_i}$ and ${\mathcal{O}}_X(-D){\mathrm{\mid }}_{U_i}=f_i{\mathcal{O}}_{U_i}$.

Dual of a local generator. For each $i$ let ${\lambda }_i\in \mathcal{H}o{m}_{{\mathcal{O}}_X}({\mathcal{O}}_X(D),{\mathcal{O}}_X)(U_i)$ be the functional determined by ${\lambda }_i(f_i^{-1})=1$; it is a basis of the free rank-one ${\mathcal{O}}_{U_i}$-module ${\mathcal{O}}_X(D{)}^{\vee }{\mathrm{\mid }}_{U_i}$ by [F3]. Hence ${\mathcal{O}}_X(D{)}^{\vee }{\mathrm{\mid }}_{U_i}={\mathcal{O}}_{U_i}\text{\hspace{0.17em}⁣}\cdot {\lambda }_i$.

The addition isomorphism. For each $i$ there is a unique ${\mathcal{O}}_{U_i}$-linear isomorphism ${\phi }_i:{\mathcal{O}}_X(D)\otimes {\mathcal{O}}_X(E){\mathrm{\mid }}_{U_i}\to {\mathcal{O}}_X(D+E){\mathrm{\mid }}_{U_i}$ sending $a(f_i^{-1}\otimes g_i^{-1})$ to $a(f_i{g}_i{)}^{-1}$, and the ${\phi }_i$ agree on overlaps and glue to a global isomorphism $\phi :{\mathcal{O}}_X(D)\otimes {\mathcal{O}}_X(E)\to {\mathcal{O}}_X(D+E)$ by [F5]. Indeed, both sides are free of rank one on $U_i$, with the displayed generators. On an overlap $U_i\cap U_j$ write $u=f_i\mathrm{/}{f}_j$ and $v=g_i\mathrm{/}{g}_j$, units of ${\mathcal{O}}_X(U_i\cap U_j)$; then $f_i^{-1}\otimes g_i^{-1}=(uv{)}^{-1}(f_j^{-1}\otimes g_j^{-1})$ and $(f_i{g}_i{)}^{-1}=(uv{)}^{-1}(f_j{g}_j{)}^{-1}$, so the transition units of source and target coincide in the displayed trivialisations and ${\phi }_i$, ${\phi }_j$ agree on the overlap. Hence the ${\phi }_i$ glue, and the glued map is an isomorphism because it is one on every chart.

The inverse isomorphism. There are unique ${\mathcal{O}}_{U_i}$-linear isomorphisms ${\tau }_i:{\mathcal{O}}_X(-D){\mathrm{\mid }}_{U_i}\to {\mathcal{O}}_X(D{)}^{\vee }{\mathrm{\mid }}_{U_i}$ sending $a{f}_i$ to $a{\lambda }_i$, where ${\lambda }_i(f_i^{-1})=1$, and these ${\tau }_i$ agree on overlaps and glue by [F5] to an isomorphism $\tau :{\mathcal{O}}_X(-D)\to {\mathcal{O}}_X(D{)}^{\vee }$. Indeed, on $U_i\cap U_j$ write $f_i=u{f}_j$ with $u=f_i\mathrm{/}{f}_j$ a unit; then the dual bases satisfy ${\lambda }_i=u{\lambda }_j$, because ${\lambda }_i(f_i^{-1})={\lambda }_i(u^{-1}{f}_j^{-1})=1$ forces ${\lambda }_i=u{\lambda }_j$. Hence $a{f}_i=au{f}_j\mapsto au{\lambda }_j=a{\lambda }_i$, so ${\tau }_i$ and ${\tau }_j$ agree on the overlap, and $\tau$ is an isomorphism because each ${\tau }_i$ carries the basis $f_i$ of ${\mathcal{O}}_X(-D){\mathrm{\mid }}_{U_i}$ to the basis ${\lambda }_i$ of ${\mathcal{O}}_X(D{)}^{\vee }{\mathrm{\mid }}_{U_i}$.

Conclusion. There are canonical isomorphisms ${\mathcal{O}}_X(D+E)\cong {\mathcal{O}}_X(D)\otimes {\mathcal{O}}_X(E)$ and ${\mathcal{O}}_X(-D)\cong {\mathcal{O}}_X(D{)}^{\vee }$; the first is characterised by $(f_i{g}_i{)}^{-1}\mapsto f_i^{-1}\otimes g_i^{-1}$ and the second by $f_i\mapsto {\lambda }_i$ with ${\lambda }_i(f_i^{-1})=1$. The second is the canonical inverse of ${\mathcal{O}}_X(D)$ described by [F4]: combining it with the first for the pair $(D,-D)$ gives ${\mathcal{O}}_X(-D)\otimes {\mathcal{O}}_X(D)\cong {\mathcal{O}}_X(0)={\mathcal{O}}_X$, the evaluation pairing. The construction uses only the given equations; no trivialisations are chosen and no choice principle is used.