Sorry for the unspecific title. Here the actual question:
Given categories $\mathcal{A},\mathcal{B}$, let $S$ be the canonical functor $\mathcal{B} \times \mathcal{A} \to \mathcal{A} \times \mathcal{B}$.
Given a bifunctor $F : \mathcal{A} \times \mathcal{B} \to \mathcal{C}$, is there an established name or notation for:
- the functor $F\circ S : \mathcal{B} \times \mathcal{A}\to \mathcal{C}$
- the "induced" functor $\mathcal{B} \to \mathcal{C}^\mathcal{A}$ (which I would call the exponential transpose of $F\circ S$, but that is unsatisfactory)
A partial answer will suffice.