Definition of $Y^f$ in category theory: exponential pullback. Pullback exponential is defined as follows.
Let $f \colon A \to B$, $g \colon X \to Y$. Let $C$ be a category with bi-closed, symmetric monoidal structure.
Then the exponential pullback is defined as follows.

It's clear to me that I obtain $g^A$ by applying the right adjoint to $A \otimes -$ to $g$. But I'm having trouble understanding what is $Y^f$. If this was in Set, it would be clear that $Y^f$ is just pre-composition, but how would I understand $Y^f$ in general category?
Thanks!

$Y^f$ is precomposition, internalized via the tensor-hom adjunction.
By the adjunction, to give an arrow $Y^B\to Y^A$, it suffices to give an arrow $Y^B\otimes A\to Y$. We have an arrow $\mathrm{eval}\colon Y^B\otimes B\to Y$. So we can get the desired arrow as $\mathrm{eval}\circ (\mathrm{id}_{Y^B}\otimes f)$. Intuitively, this says "apply $f$ to the input in $A$ before evaluating the map in $Y^B$".
Similarly, $g^A$ can be viewed as internalized postcomposition. By the adjunction, to give an arrow $X^A\to Y^A$, it suffices to give an arrow $X^A\otimes A\to Y$. We have an arrow $\mathrm{eval}\colon X^A\otimes A\to X$. So we can get the desired arrow as $g\circ \mathrm{eval}$. Intuitively, this says "apply $g$ to the output in $X$ after evaluating the map in $X^A$".
