Left adjoint and right adjoint/ Nakayama isomorphism

You should learn something about the representation theory of finite groups. It suffices to understand the case that $X, Y$ have one object (note that the definition is as a direct sum of things that happen at one object); call those objects $x, y$ and let $G = \text{Aut}(x), H = \text{Aut}(y)$. Then $f$ may be regarded as a morphism $f : H \to G$. The functor categories $[X, \text{Vect}]$ and $[Y, \text{Vect}]$ are precisely the categories $\text{Rep}(G), \text{Rep}(H)$ of representations of $G$ and $H$ respectively. The pullback functor

$$f^{\ast} : \text{Rep}(G) \to \text{Rep}(H)$$

is known in representation theory as restriction and commonly denoted $\text{Res}_H^G$. Restriction happens to have two equivalent descriptions, and it will be cleaner for me to describe these by working in slightly greater generality. Namely, let $f : R \to S$ be a homomorphism of rings. $f$ induces a pullback functor

$$\text{Res}_R^S : S\text{-Mod} \to R\text{-Mod}$$

from the category of left $S$-modules to the category of left $R$-modules. (Recall that the category of left $R$-modules is the functor category $[R, \text{Ab}]$). When $R = \mathbb{C}[H], S = \mathbb{C}[G]$ are group algebras we recover the picture above.

Restriction can be described in two ways. The first description is (writing $_RM_S$ for an $(R, S)$-bimodule $M$)

$$_S M \mapsto \text{Hom}_S(_S S_R, _SM)$$

where $_S S_R$ is $S$ regarded as a left $S$-module in the obvious way and as a right $R$-module via the map $f$ above, and

$$_S M \mapsto _R S_S \otimes_S M$$

where $_R S_S$ is $S$ regarded as a right $S$-module in the obvious way and as a left $R$-module via the map $f$ above. Via the tensor-hom adjunction, this shows that restriction has both a left and a right adjoint. The left adjoint is called extension of scalars in module theory and induction in group theory and is given by

$$_R N \mapsto _S S_R \otimes_R N.$$

In group theory it is usually denoted $\text{Ind}_H^G$.

I don't know if the right adjoint has a standard name in module theory. In group theory I think it is sometimes called coinduction (so maybe in module theory it should be called coextension?) and is given by

$$_R N \mapsto \text{Hom}_R(_R S_S, _R N).$$

For an arbitrary map $f : R \to S$ of rings, there is no reason to expect these two functors to be naturally isomorphic. However, if $f : \mathbb{C}[H] \to \mathbb{C}[G]$ is a map on group algebras induced by a group homomorphism, then induction and coinduction are naturally isomorphic (which is why you won't see textbooks on the representation theory of finite groups talk about coinduction). The explicit isomorphism between them is what Morton calls the Nakayama isomorphism; it is an isomorphism

$$\text{Nak} : \text{Hom}_{\mathbb{C}[H]}(\mathbb{C}[G], V) \to \mathbb{C}[G] \otimes_{\mathbb{C}[H]} V$$

(I am dropping the bimodule subscripts for ease of notation) given by

$$\text{Hom}_{\mathbb{C}[H]}(\mathbb{C}[G], V) \ni \phi \mapsto \frac{1}{|H|} \sum_{g \in G} g^{-1} \otimes \phi(g) \in \mathbb{C}[G] \otimes_{\mathbb{C}[H]} V.$$

From a representation-theoretic perspective, the existence of this isomorphism merely reflects that $\mathbb{C}[G]$, as a representation of $H$, is self-dual, and this in turn follows from the fact that it has real character. But the Nakayama isomorphism is a preferred isomorphism for reasons I do not completely understand.