  Motivated by problems in categorical database theory, we introduce and
compare two notions of presentation for profunctors, uncurried and curried,
which arise intuitively from thinking of profunctors either as functors
$\mathcal{C}^\text{op} \times \mathcal{D} \to \textbf{Set}$ or
$\mathcal{C}^\text{op} \to \textbf{Set}^{\mathcal{D}}$. Although the Cartesian
closure of $\textbf{Cat}$ means these two perspectives can be used
interchangeably at the semantic level, a surprising amount of subtlety is
revealed when looking through the lens of syntax. Indeed, we prove that finite
uncurried presentations are strictly more expressive than finite curried
presentations, hence the two notions do not induce the same class of finitely
presentable profunctors. Moreover, an explicit construction for the composite
of two curried presentations shows that the class of finitely curried
presentable profunctors is closed under composition, in contrast with the
larger class of finitely uncurried presentable profunctors, which is not. This
shows that curried profunctor presentations are more appropriate for
computational tasks that use profunctor composition. We package our results on
curried profunctor presentations into a double equivalence from a syntactic
double category into the double category of profunctors. Finally, we study the
relationship between curried and uncurried presentations, leading to the
introduction of curryable presentations. These constitute a subcategory of
uncurried presentations which is equivalent to the category of curried
presentations, therefore acting as a bridge between the two syntactic choices.
