  We discuss the following problem: how can an arbitrary Fourier-Mukai
transform $\phi: \mathrm{D}^{\mathrm{b}}( \mathbb{P}^a ) \rightarrow
\mathrm{D}^{\mathrm{b}}( \mathbb{P}^b )$ between the bounded derived categories
of two projective spaces of dimensions $a$ and $b$ be expressed in explicit
terms as an exact functor between the homotopy categories
$\mathrm{K}^{\mathrm{b}}( \mathbb{B}^a ) \rightarrow \mathrm{K}^{\mathrm{b}}(
\mathbb{B}^b )$ generated by the full strong exceptional sequences of the line
bundles $\mathbb{B}^a = \{\mathcal{O}(-a), \dots, \mathcal{O}\}$ and
$\mathbb{B}^b = \{\mathcal{O}(-b), \dots, \mathcal{O}\}$? We show that this
problem can be reduced to the following task which is independent of any
prescribed Fourier-Mukai kernel: finding an $A_{\infty}$-functor $P$ in
explicit terms whose induced functor on homotopy categories yields the
embedding of $\{ \mathcal{O}(i) \boxtimes \mathcal{O}(j) \mid i = -2a, \dots,
0, ~~j = -2b, \dots 0 \}$ into $\mathrm{D}^{\mathrm{b}}( \mathbb{P}^{a} \times
\mathbb{P}^{b})$.
  As our main technical tool we provide an explicit formula for the lift of an
$A_{\infty}$-functor $F: \mathbf{A} \rightarrow \mathbf{B}$ between a dg
category $\mathbf{A}$ and a pretriangulated dg category $\mathbf{B}$ to the
pretriangulated hull of $\mathbf{A}$ given by the universal property of
pretriangulated hulls. As a further application of this tool, we provide a
simple example of two non-isomorphic exact functors between triangulated
categories that coincide on the full subcategory generated by a full strong
exceptional sequence.
