I am reading a paper "2-vector spaces and groupoid" by Jeffrey Morton and I need a help to understand the following.
Let $X$ and $Y$ be finite groupoids. Let $[X, \mathbb{Vect}]$ be a functor category from $X$ to $\mathbb{Vect}$. Suppose we have a functor $f: Y \rightarrow X$. The author defines a pushforward $f_{*} :[Y, \mathbb{Vect}] \rightarrow [X, \mathbb{Vect}]$ as follows.
For each object $x \in X$, the comma category $(f \downarrow x)$ has object which are objects $y \in Y$ with maps $f(y) \rightarrow x$ in $X$, and morphisms which are morphisms $a: y \rightarrow y'$ whose images make the evident triangle in $X$ commute.
Then the author defines for each $F\in [Y, \mathbb{Vect}]$, $f_{*}(F)(x):=colim F(f \downarrow x)$.
He also shows that $f_{*}(F)(y)=\bigoplus_{f(x)\cong y} \mathbb{C}[Aut(y)]\otimes_{\mathbb{C}[Aut(x)]}F(x)$ and fimilarly calculate $f^*f_*$ and $f_*f^*$.
Later he says that this description accords with the susal description of these functors in the left adjunction. Then he mentions that the right adjoint is given as
$f_*F(x)= \bigoplus_{[y], f(y)\cong x}hom_{\mathbb{C}[Aut(x)]}(\mathbb{C}[Aut(y), F(y)).$
Then we says:
The Nalayama isomorphism gives the duality between the two descriptions of $f_*$, in terms of $hom_{\mathbb{C}[Aut(x)]}$ and $\otimes_{\mathbb{C}[Aut(x)]}$ by means of the exterior trace map.
I understood the calculation of $f_*F(x)$ using a colimit defined above. But I don't know what it means by saying left or right adjoint. Also I don't know how to get the formula for the right adjoint. I even don't know what is Nakayama isomorphism. I searched for it but I couldn't find a good resource.
So I would like to know what is going on here. Especially, I'd like to know;
Why is the first construction using a colimit called "left adjoint"?
What is the right adjoint and how is it defined and how to calculate it to get the formula above.
What is Nakayama isomorphism?
I have never studied these things so I don't know where to look up. I also want good references.
I appreciate any help. Thank you in advance.