I'm reading these lecture notes.
On p.18, $\widetilde FA$ is supposed to be a functor from $\mathbb B$ to $\mathbb C$. To define such a functor, one must say how it acts on objects and on morphisms of $\mathbb B$. However, the notes only say how it acts on objects. How does it act on morphisms?
Similarly, on p.20, $yA$ is supposed to be a functor, but it is not specified how it acts on morphisms.
Moreover, on p.18 given a functor $F:\mathbb A\times \mathbb B\to \mathbb C$ the transpose of $F$ is defined as a functor $\widetilde F:\mathbb A\to \mathbb C^\mathbb B$. But on p.20 it is said that the transpose of $\mathcal C^{op}\times\mathcal C\to \mathcal{Set}$ is a functor $\mathcal C\to \mathcal {Set}^{\mathcal C^{op}}$, contradicting the previous definition on p.18... If there are two notions of a transpose of a functor, what is the definition for the general case that specializes to what is written on p.20? (The one on p.18 doesn't!)
Given $f:B\to B'$, $\tilde{F}A(f)$ is just the morphism $F(id_A,f)$ in $\mathbb{C}$. In other words, $\tilde{F}A$ just fixes the $A$ argument of $F$, and treats it as a functor that only varies in the $B$ variable.
As for the discrepency in the definition of transpose, $F:A\times B\to C$ can be transposed to a functor $A\to C^B$ or a functor $B\to C^A$. If we take only the first case as the "official" one, we can get the second simply by composing $F:A\times B\to C$ with the twist map $(X,Y)\mapsto(Y,X):B\times A\to A\times B$ and taking the transpose of the composite.