I'm having huge trouble with the exercise 3.1.xii from the book Category theory in contexts:
I.e. prove that the category $\int \text{Cone(-,F)}$ is equivalent to $\int \text{Cone(-,FE)}$, where $\text{Cone(-,G)}$ sends an object $c \in C$ to set of cones over $G$ with apex $c$ and $\int(-)$ denotes category of elements.
Assume $E':J \rightarrow I$ is an equivalence such that $EE'$, $E'E$ are naturally isomorphic to identities.
First I have defined a functor $H:\int \text{Cone(-,F)}\rightarrow\int \text{Cone(-,FE)}$ that takes natural transformation $(\lambda _{j}:c\rightarrow Fj)$ to transformation $(\lambda_{Ei}:c\rightarrow FEi)$. Then I defined composite functor $K:\int \text{Cone(-,FE)} \rightarrow \int \text{Cone(-,F)}$ that first takes natural transformation $\mu = (\mu_{i}:c \rightarrow FEi)$ to $\mu E' = (\mu_{E'j}:c\rightarrow FEE'j)$ and then (assuming $\beta:EE' \Rightarrow 1_{J}$ is natural iso) to $(\beta_{j}\mu_{E'j}:c\rightarrow Fj)$.
According to my computations $KH, HK$ are identities (so the categories are isomorphic) - hence my assumption that my "proof" is wrong.
Could someone point out why is my approach wrong and suggest better one?
Thanks.