= Solution
{c}
The <tensor product of sheaves> is the <sheafification> of the presheaf
$$
U\longmapsto\mathcal F(U)\otimes_{\mathcal O_X(U)}\mathcal G(U),
$$
with 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 map
$$
(\mathcal F\otimes_{\mathcal O_X}\mathcal G)_P
\longrightarrow\mathcal F_P\otimes_{\mathcal O_{X,P}}\mathcal G_P.
$$
It 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> $\phi^{-1}\mathcal F$: sheafify $U\mapsto\operatorname*{colim}_{V\supseteq\phi(U)}\mathcal F(V)$. The morphism of <sheaves of rings> $\phi^{-1}\mathcal O_Y\to\mathcal O_X$ then defines
$$
\phi^*\mathcal F=\mathcal O_X\otimes_{\phi^{-1}\mathcal O_Y}\phi^{-1}\mathcal F,
\qquad
(\phi^*\mathcal F)_P\cong\mathcal O_{X,P}\otimes_{\mathcal O_{Y,\phi(P)}}\mathcal F_{\phi(P)}.
$$
Exactness 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 $\mathcal O_{Y,\phi(P)}$ gives the exact stalk sequence of the pullbacks. Thus $\phi^*$ preserves sequences ending in zero, as required.
A <quasi-coherent sheaf> is a <sheaf of modules> locally of the form $\widetilde N$ 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 $\widetilde N(D(b))=N_b$, using <localization of a module>. Suppose $\mathcal F|_V\cong\widetilde N$ and $U\subseteq\phi^{-1}V$ is affine, with ring $C$. The induced algebra map $B\to C$ gives a canonical map
$$
\widetilde{C\otimes_BN}\longrightarrow(\phi^*\mathcal F)|_U.
$$
At a point $P\in U$, both sides identify with $\mathcal O_{X,P}\otimes_BN$: on the right use $\mathcal F_{\phi(P)}\cong\mathcal O_{Y,\phi(P)}\otimes_BN$ 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 value
$$
M\otimes_R B_b\cong(M\otimes_RB)_b.
$$
Consequently $M_Y|_V\cong\widetilde{M\otimes_RB}$, 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 $\phi^*M_Y\to M_X$. At $P$, it is the associativity isomorphism
$$
\mathcal O_{X,P}\otimes_{\mathcal O_{Y,\phi(P)}}
\bigl(M\otimes_R\mathcal O_{Y,\phi(P)}\bigr)
\cong M\otimes_R\mathcal O_{X,P}.
$$
It is therefore a <sheaf of modules> isomorphism. In particular,
$$
\boxed{(\mathcal F\otimes\mathcal G)_P\cong\mathcal F_P\otimes_{\mathcal O_{X,P}}\mathcal G_P,\qquad M_X\cong\phi^*M_Y.}
$$
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