Different definitions of profunctors

Question

In nlab a profunctor from category $\mathbb{C}$ to category $\mathbb{D}$ is defined to be a functor $\mathbb{D}^{\text{op}} \times \mathbb{C} \overset{F}{\rightarrow} \textbf{Set}.$ However in Fong and Spivak (Definition 4.2) a profunctor from category $\mathbb{C}$ to category $\mathbb{D}$ is defined to be a functor $\mathbb{C}^{\text{op}} \times \mathbb{D} \overset{G}{\rightarrow} \textbf{Set}.$ I would like to know which of these definitions is most common (especially for discussing the category of profunctors), and I would like to know how to convert between "profunctors" defined according to these two different definitions.