A construction of a pushforward of Vect-presheves. 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 $f: X \rightarrow Y$. We want to construct a pushforward $f_{*} :[X, \mathbb{Vect}] \rightarrow [Y, \mathbb{Vect}]$.
For each object $y \in Y$, the comma category $(f \downarrow y)$ has object which are objects $x \in X$ with maps $f(x) \rightarrow y$ in $Y$, and morphisms which are morphisms $a: x \rightarrow x'$ whose images make the evident triangle in $Y$ commute.
Then the author defines for each $F\in [X, \mathbb{Vect}]$, $f_{*}(F)(y):=colim F(f \downarrow y)$.
He also shows that $f_{*}(F)(y)=\bigoplus_{f(x)\cong y} \mathbb{C}[Aut(y)]\otimes_{\mathbb{C}[Aut(x)]}F(x)$.
The first thing I don't understand is the definition of $ colim F(f \downarrow y)$.
Is it a colimit of a functor $F: (f \downarrow y) \rightarrow \mathbb{Vect}$ by regarding the comma category as a subcategory of $X$?
The second thing I don't understand is the proof of the explicit formula above. The author first consider the case when $X$ and $Y$ are just finite groups $G$ and $H$ respectively and $f :G \rightarrow H$ is a homomorphism and we have a representation of $G$ on a vector space $V$. In this case we have $f_{*}V=\mathbb{C}[H] \otimes_{\mathbb{C}[G]} V$. 
I don't know how to calculate the colimit above and deduce this formula.
I appreciate any help. Thank you in advance.
 A: The answer to your first question is roughly yes, although in general a slice category may not be a subcategory since the image of an object in the source can admit many maps to the target, and these all give distinct (possibly non-isomorphic) objects in the slice.  Of course, you can still define the pullback along the forgetful functor.
As for your second question, this is just the fact that the "tensoring up" functor $\mathbb{C}[H] \otimes_{\mathbb{C}[G]} - :\mbox{Rep}(G) \rightarrow \mbox{Rep}(H)$ is adjoint to the forgetful (or "restriction of structure") functor $U:\mbox{Rep}(H) \rightarrow \mbox{Rep}(G)$.
To actually compute this colimit, let's write $\mathcal{G}$ and $\mathcal{H}$ for the categories, $*_G$ and $*_H$ for their unique objects, and identify the morphisms with the sets $G$ and $H$.  Then the objects of $(f\downarrow *_H)$ are given by pairs $(*_G,h:*_H \rightarrow *_H)$ (for all $h\in H$), and the morphisms of $(f\downarrow *_H)$ are given by
$$ \mbox{Hom}_{(f\downarrow *_H)}((*_G,h_1:*_H \rightarrow *_H),(*_G,h_2:*_H \rightarrow *_H) = \{g\in G : f(g)=h_2^{-1}h_1\} $$
(using the standard convention for composition).
Now we can explicitly describe the diagram $(f \downarrow *_H) \xrightarrow{U} \mathcal{G} \xrightarrow{F} \mbox{Vect}$: it takes every object to $V$, and to the morphism $f(g)=h_2^{-1}h_1$ it associates the linear homomorphism (actually isomorphism) $F(g)$ from the copy of $V$ over $h_1$ to the copy of $V$ over $h_2$.
Colimits are built as follows: given a diagram $D:\mathcal{I} \rightarrow \mathcal{C}$,
$$ \mbox{colim}(D) \cong \mbox{coeq} \left( \coprod_{(i_0 \rightarrow i_1)\in \mbox{mor}(\mathcal{I})} D(i_0) \rightrightarrows \coprod_{j \in \mbox{ob}(\mathcal{I})} D(j) \right),$$
where one map takes the $D(i_0)$ over $(i_0 \rightarrow i_1)$ to $D(i_1)$ over $j=i_1$ via $D(i_0 \rightarrow i_1)$, and the other map takes $D(i_0)$ over $(i_0 \rightarrow i_1)$ to $D(i_0)$ over $j=i_0$ via the identity map.  Of course, in $\mbox{Vect}$ all this means is that you take a big direct sum of all the vector spaces in the diagram and then take the quotient where we identify $v\in D(i_0)$ with $(D(i_0 \rightarrow i_1))(v) \in D(i_1)$ (for all $(i_0 \rightarrow i_1) \in \mbox{mor}(\mathcal{I})$).
So for our situation, we start with the direct sum of a bunch of copies of $V$ indexed by $H$, i.e. $\bigoplus_H V \cong \mathbb{C}[H] \otimes V$.  (Categorically, one would say that $\mbox{Vect}$ is tensored (or copowered) over $\mbox{Set}$, which essentially means that we have a reasonable and (bi)functorial notion of "taking coproducts of an object indexed by a set".)  Then, if we have a vector $v\in V$ considered as sitting over $h_1$ -- that is, the element $h_1 \otimes v$ -- and $f(g)=h_2^{-1}h_1$ is a morphism from $h_1$ to $h_2$, then we make the identification $h_1 \otimes v \sim h_2 \otimes g \cdot v$.  But notice that this is just the usual definition of trading off the right action of $\mathbb{C}[G]$ on $\mathbb{C}[H]$ via $f$.  (To be precise, this action comes from the functoriality and the product-preservation of the tensored structure I mentioned above.)  So finally, $\mbox{colim}_{(f \downarrow *_H)} F \circ U \cong \mathbb{C}[H] \otimes_{\mathbb{C}[G]} V$.
