In response to David's comment below:
Modulo some details, a planar algebra is equivalent to a pivotal 2-category whose 2-morphisms are vector spaces and whose 1-morphisms are finitely generated. The standard example is constructed from a pair of factors (irreducible von Neumann algebras) $N\subset M$. From this data we construct a 2-category whose objects are $N$ and $M$, whose 1-morphisms are generated by the two bimodules $_N M_M$ and $_M M_N$, and whose 2-morphisms are intertwinors.
(So for example the 1-morphisms are $M\otimes_N M\otimes_N\cdots\otimes_N M$, thought of as either an $N$-$N$ or $N$-$M$ or $M$-$N$ or $M$-$M$ bimodule.) You can think of the usual planar algebra definition as axiomatizing the "string diagrams" you would draw for this 2-category.
The diagrams in the paper I referred to are rotated 90 degrees from my explanation above. The left and right sides of the rectangles in the paper correspond to the $f$ and $f'$ of your (David's) original question. The tops of the rectangles corresponds to your $\phi$, and the interiors of the rectangles correspond to your $\phi^\sharp$.