By Lemma \ref{lemma-duality-at-point} we conclude that for every coherent $\mathcal{O}_X$-module $\mathcal{F}$ whose support is a closed point there are isomorphisms $$ H^0(X, M \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{F}) = H^0(X, F(M) \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{F}) $$ functorial in $M$. \medskip\noindent Let $x \in X$ be a closed point and apply the above with $\mathcal{F} = \mathcal{O}_x$ the skyscraper sheaf with value $\kappa(x)$ at $x$. We find $$ \dim_{\kappa(x)}  \text{Tor}^{\mathcal{O}_{X, x}}_p(M_x, \kappa(x)) = \dim_{\kappa(x)}  \text{Tor}^{\mathcal{O}_{X, x}}_p(F(M)_x, \kappa(x)) $$ for all $p \in \mathbf{Z}$. In particular, if $H^i(M) = 0$ for $i > 0$, then $H^i(F(M)) = 0$ for $i > 0$ by Lemma \ref{lemma-orthogonal-point-sheaf}. \medskip\noindent If $\mathcal{E}$ is locally free of rank $r$, then $F(\mathcal{E})$ is locally free of rank $r$. This is true because a perfect complex $K$ over $\mathcal{O}_{X, x}$ with $$ \dim_{\kappa(x)} \text{Tor}^{\mathcal{O}_{X, x}}_i(K, \kappa(x)) = \left\{ \begin{matrix} r & \text{if} & i = 0 \\ 0 & \text{if} & i \not = 0 \end{matrix} \right. $$ is equal to a free module of rank $r$ placed in degree $0$. See for example More on Algebra, Lemma \ref{more-algebra-lemma-lift-perfect-from-residue-field}. \medskip\noindent If $M$ is supported on a closed subscheme $Z \subset X$, then $F(M)$ is also supported on $Z$. This is clear because we will have $M \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{O}_x = 0$ for $x \not \in Z$ and hence the same will be true for $F(M)$ and hence we get the conclusion from Lemma \ref{lemma-orthogonal-point-sheaf}. \medskip\noindent In particular $F(\mathcal{O}_x)$ is supported at $\{x\}$. Let $i \in \mathbf{Z}$ be the minimal integer such that $H^i(\mathcal{O}_x) \not = 0$. We know that $i \leq 0$. If $i < 0$, then there is a morphism $\mathcal{O}_x[-i] \to F(\mathcal{O}_x)$ which contradicts the fact that all morphisms $\mathcal{O}_x[-i] \to \mathcal{O}_x$ are zero. Thus $F(\mathcal{O}_x) = \mathcal{H}[0]$ where $\mathcal{H}$ is a skyscraper sheaf at $x$. \medskip\noindent Let $\mathcal{G}$ be a coherent $\mathcal{O}_X$-module with $\dim(\text{Supp}(\mathcal{G})) = 0$. Then there exists a filtration $$ 0 = \mathcal{G}_0 \subset \mathcal{G}_1 \subset \ldots \subset \mathcal{G}_n = \mathcal{G} $$ such that for $n \geq i \geq 1$ the quotient $\mathcal{G}_i/\mathcal{G}_{i - 1}$ is isomorphic to $\mathcal{O}_{x_i}$ for some closed point $x_i \in X$. Then we get distinguished triangles $$ F(\mathcal{G}_{i - 1}) \to F(\mathcal{G}_i) \to F(\mathcal{O}_{x_i}) $$ and using induction we find that $F(\mathcal{G}_i)$ is a coherent sheaf placed in degree $0$. \medskip\noindent Let $\mathcal{G}$ be a coherent $\mathcal{O}_X$-module. We know that $H^i(F(\mathcal{G})) = 0$ for $i > 0$. To get a contradiction assume that $H^i(F(\mathcal{G}))$ is nonzero for some $i < 0$. We choose $i$ minimal with this property so that we have a morphism $H^i(F(\mathcal{G}))[-i] \to F(\mathcal{G})$ in $D_{perf}(\mathcal{O}_X)$. Choose a closed point $x \in X$ in the support of $H^i(F(\mathcal{G}))$. By More on Algebra, Lemma \ref{more-algebra-lemma-kollar-kovacs-pseudo-coherent} there exists an $n > 0$ such that $$ H^i(F(\mathcal{G}))_x \otimes_{\mathcal{O}_{X, x}} \mathcal{O}_{X, x}/\mathfrak m_x^n \longrightarrow \text{Tor}^{\mathcal{O}_{X, x}}_{-i}(F(\mathcal{G})_x, \mathcal{O}_{X, x}/\mathfrak m_x^n) $$ is nonzero. Next, we take $m \geq 1$ and we consider the short exact sequence $$ 0 \to \mathfrak m_x^m \mathcal{G} \to \mathcal{G} \to \mathcal{G}/\mathfrak m_x^m\mathcal{G} \to 0 $$ By the above we know that $F(\mathcal{G}/\mathfrak m_x^m\mathcal{G})$ is a sheaf placed in degree $0$. Hence $H^i(F(\mathfrak m_x^m \mathcal{G})) \to H^i(F(\mathcal{G}))$ is an isomorphism. Consider the commutative diagram $$ \xymatrix{ H^i(F(\mathfrak m_x^m\mathcal{G}))_x \otimes_{\mathcal{O}_{X, x}} \mathcal{O}_{X, x}/\mathfrak m_x^n \ar[r] \ar[d] & \text{Tor}^{\mathcal{O}_{X, x}}_{-i}(F(\mathfrak m_x^m\mathcal{G})_x, \mathcal{O}_{X, x}/\mathfrak m_x^n) \ar[d] \\ H^i(F(\mathcal{G}))_x \otimes_{\mathcal{O}_{X, x}} \mathcal{O}_{X, x}/\mathfrak m_x^n \ar[r] & \text{Tor}^{\mathcal{O}_{X, x}}_{-i}(F(\mathcal{G})_x, \mathcal{O}_{X, x}/\mathfrak m_x^n) } $$ Since the left vertical arrow is an isomorphism and the bottom arrow is nonzero, we conclude that the right vertical arrow is nonzero for all $m \geq 1$. On the other hand, by the first paragraph of the proof, we know this arrow is isomorphic to the arrow $$ \text{Tor}^{\mathcal{O}_{X, x}}_{-i}(\mathfrak m_x^m\mathcal{G}_x, \mathcal{O}_{X, x}/\mathfrak m_x^n) \longrightarrow \text{Tor}^{\mathcal{O}_{X, x}}_{-i}(\mathcal{G}_x, \mathcal{O}_{X, x}/\mathfrak m_x^n) $$ However, this arrow is zero for $m \gg n$ by More on Algebra, Lemma \ref{more-algebra-lemma-tor-annihilated} which is the contradiction we're looking for. \medskip\noindent Thus we know that $F$ preserves coherent modules. By Lemma \ref{lemma-exact-functor-preserving-Coh} we find $F$ is a sibling to the Fourier-Mukai functor $F'$ given by a coherent $\mathcal{O}_{X \times X}$-module $\mathcal{K}$ flat over $X$ via $\text{pr}_1$ and finite over $X$ via $\text{pr}_2$. Since $F(\mathcal{O}_X)$ is an invertible $\mathcal{O}_X$-module $\mathcal{L}$ placed in degree $0$ we see that $$ \mathcal{L} \cong F(\mathcal{O}_X) \cong F'(\mathcal{O}_X) \cong \text{pr}_{2, *}\mathcal{K} $$ Thus by Lemma \ref{lemma-pushforward-invertible-pre} there is a morphism $s : X \to X \times X$ with $\text{pr}_2 \circ s = \text{id}_X$ such that $\mathcal{K} = s_*\mathcal{L}$. Set $f = \text{pr}_1 \circ s$. Then we have \begin{align*} F'(M) & = R\text{pr}_{2, *}(L\text{pr}_1^*K \otimes \mathcal{K}) \\ & = R\text{pr}_{2, *}(L\text{pr}_1^*M \otimes s_*\mathcal{L}) \\ & = R\text{pr}_{2, *}(Rs_*(Lf^*M \otimes \mathcal{L})) \\ & = Lf^*M \otimes \mathcal{L} \end{align*} where we have used Derived Categories of Schemes, Lemma \ref{perfect-lemma-cohomology-base-change} in the third step. Since for all closed points $x \in X$ the module $F(\mathcal{O}_x)$ is supported at $x$, we see that $f$ induces the identity on the underlying topological space of $X$. We still have to show that $f$ is an isomorphism which we will do in the next paragraph. \medskip\noindent Let $x \in X$ be a closed point. For $n \geq 1$ denote $\mathcal{O}_{x, n}$ the skyscaper sheaf at $x$ with value $\mathcal{O}_{X, x}/\mathfrak m_x^n$. We have $$ \Hom_X(\mathcal{O}_{x, m}, \mathcal{O}_{x, n}) \cong \Hom_X(\mathcal{O}_{x, m}, F(\mathcal{O}_{x, n})) \cong \Hom_X(\mathcal{O}_{x, m}, f^*\mathcal{O}_{x, n} \otimes \mathcal{L}) $$ functorially with respect to $\mathcal{O}_X$-module homomorphisms between the $\mathcal{O}_{x, n}$. (The first isomorphism exists by assumption and the second isomorphism because $F$ and $F'$ are siblings.) For $m \geq n$ we have $\mathcal{O}_{X, x}/\mathfrak m^n = \Hom_X(\mathcal{O}_{x, m}, \mathcal{O}_{x, n})$ via the action on $\mathcal{O}_{x, n}$ we conclude that $f^\sharp : \mathcal{O}_{X, x}/\mathfrak m_x^n \to \mathcal{O}_{X, x}/\mathfrak m_x^n$ is bijective for all $n$. Thus $f$ induces isomorphisms on complete local rings at closed points and hence is \'etale (\'Etale Morphisms, Lemma \ref{etale-lemma-characterize-etale-completions}). Looking at closed points we see that $\Delta_f : X \to X \times_{f, X, f} X$ (which is an open immersion as $f$ is \'etale) is bijective hence an isomorphism. Hence $f$ is a monomorphism. Finally, we conclude $f$ is an isomorphism as Descent, Lemma \ref{descent-lemma-flat-surjective-quasi-compact-monomorphism-isomorphism} tells us it is an open immersion.