= Solution
{c}
\b[Constructing the subsheaf.] On an <integral scheme>, every nonempty <open subset> is an <irreducible topological space> and hence a <connected space>. A <locally constant function> from such an <open subset> to $F$ is constant. Thus the <constant sheaf> $\underline F$ has sections $F$ on every nonempty <open subset>, and the <structure sheaf> embeds in it by restriction to the <generic point>. Empty members of the given <open cover> may be discarded.
For every <open subset> $W\subseteq X$, define
$$
\mathcal L(W)=\{s\in\underline F(W):\ f_i s|_{W\cap U_i}\in\mathcal O_X(W\cap U_i)\text{ for every }i\}.
$$
This is a subsheaf of the <sheaf of modules> $\underline F$: its condition is local, and compatible sections glue in $\underline F$, with the regularity conditions gluing in $\mathcal O_X$. On $U_i\cap U_j$, the hypothesis says that $f_j/f_i$ is a section of the <sheaf of units of the structure sheaf>. Therefore multiplication by it preserves $\mathcal O_X$, and
$$
f_i^{-1}\mathcal O_X|_{U_i\cap U_j}=f_j^{-1}\mathcal O_X|_{U_i\cap U_j}.
$$
It follows that $\mathcal L|_{U_i}=f_i^{-1}\mathcal O_{U_i}$: the condition with index $i$ implies every other condition on an overlap, and the converse is part of the definition. Multiplication by $1/f_i$ is an <isomorphism> $\mathcal O_{U_i}\to\mathcal L|_{U_i}$, with inverse multiplication by $f_i$. Hence $\mathcal L$ is an <invertible sheaf>, or equivalently a <line bundle>, and
$$
\boxed{\mathcal L(U_i)=\mathcal O_X(U_i)\cdot\frac1{f_i}\subseteq F.}
$$
These local descriptions also give uniqueness as a subsheaf of $\underline F$. This construction is the <divisor line bundle> for the local rational equations $f_i$.
\b[The tensor product.] Refine the two <open covers> to $W_{ij}=U_i\cap V_j$. The ratios
$$
\frac{f_i g_j}{f_{i'}g_{j'}}=\frac{f_i}{f_{i'}}\frac{g_j}{g_{j'}}
$$
are sections of the <sheaf of units of the structure sheaf> on the corresponding overlaps, so the products give a <line bundle> $\mathcal M\subseteq\underline F$ by the construction just proved. Multiplication in $F$ defines a <sheaf morphism>
$$
\mathcal L\otimes_{\mathcal O_X}\mathcal L'\longrightarrow\mathcal M.
$$
On each $W_{ij}$ it sends the local generator $(1/f_i)\otimes(1/g_j)$ to the local generator $1/(f_i g_j)$. It is thus an <isomorphism> on an <open cover>, and hence an <isomorphism> globally. In particular,
$$
\boxed{(U_i\cap V_j,f_i g_j)\text{ represents }\mathcal L\otimes_{\mathcal O_X}\mathcal L'.}
$$
\b[The sign for projective twists.] Write <homogeneous coordinates> $[x_0:\cdots:x_n]$ on <projective space> $\mathbb P^n_{\mathbb Z}$, and let $U_i=D_+(x_i)$ be its standard <affine open cover>. For any <integer> $m$, take the nonzero elements of the <function field>
$$
\boxed{f_i=\left(\frac{x_0}{x_i}\right)^m\in\mathbb Q(x_1/x_0,\ldots,x_n/x_0)^*,\qquad 0\leq i\leq n.}
$$
Their ratios are $f_i/f_j=(x_j/x_i)^m$, which are sections of the <sheaf of units of the structure sheaf> on $U_i\cap U_j$, including when $m<0$. The <twisting sheaf on projective space> $\mathcal O(m)$ has local generator $x_i^m$ on $U_i$. Divide a rational section of $\mathcal O(m)$ by $x_0^m$ to embed it in the <constant sheaf> of the <function field>; the local generator becomes $(x_i/x_0)^m=1/f_i$. This identifies the constructed <line bundle> with $\mathcal O(m)$ and fixes the sign explicitly.
For $n\geq1$, $\operatorname{Pic}(\mathbb P^n_{\mathbb Z})\cong\mathbb Z$, with generator $[\mathcal O(1)]$; the displayed data gives all its powers, with $\mathcal O(m)\otimes\mathcal O(r)\cong\mathcal O(m+r)$ by the product rule above. If $n=0$, <projective space> is $\operatorname{Spec}\mathbb Z$, its <Picard group> is zero, and the same formula gives only the trivial <line bundle>.
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