Lemma 56.16.2. Let $k$ be a field. Let $X$ and $Y$ be smooth proper schemes over $k$. Given a $k$-linear, exact, fully faithful functor $F : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y)$ there exists a Fourier-Mukai functor $F' : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y)$ whose kernel is in $D_{perf}(\mathcal{O}_{X \times Y})$ which is a sibling to $F$.

**Proof.**
Apply Lemma 56.15.7 to $F$ and the functor $G$ constructed above.
$\square$

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