Remark [efr-YKR3]

Of course, there is an equally good "pull-back" operation on selection relations given by \{(x,kf) \mid (fx, k) \in \epsilon \} (this is the pushforward in \mathbb {S}_{\mathcal {C}^\mathrm {op}}). These are adjoint representatives of the same profunctor, which should arguably be regarded as the primary object of interest---that is, we could work with a functor \mathcal {C} \to \mathsf {\mathbb Cat}, the category of categories and profunctors, where we say two selection relations \epsilon \in \mathbb {S}_X, \epsilon ' \in \mathbb {S}_Y are related by f :X \to Y if \epsilon (x, kf) \Rightarrow \epsilon '(fx, k).

However, we will stick with the pushforward definition for now, since it is conceptually simpler and good enough for our purposes.