I've still got a problem with this exercise:
I've posted question with my old approach, which didn't unforunately get any answers. I have then come up (after getting a huge hint) with different and simpler approach and according to Giorgio Mossa's advice on the old thread, I will post it with all the details and hope it won't get marked as spam.
Category of cones over $E$ is a category whose objects are cones with apexes $c\in C$ over $F$, i.e. natural transformations $\lambda: c\Rightarrow F$, where $c$ is regarded as constant functor. Morphism between cones $\lambda : c \Rightarrow F$, $\mu : d \Rightarrow F$ is a morfism $f:c \to d \in C$ such that $\mu_{i}f=\lambda_{i}$ for all $i \in I$.
First, it is known that if $E:I \to J$ defines an equivalence of categories, then $E^*:C^J \to C^I$ (defined as $F \mapsto FE$) defines equivalence of categories of functors from $J,I$.
Because $E^{*}$ is fully faithful, there is bijection between cones $\lambda :c\Rightarrow F$ and cones $\lambda E: cE \Rightarrow FE$ (where $(\lambda E)_{i} = \lambda_{Ei}$)... these are exactly the cones $\mu:c \Rightarrow FE$, i.e. cones over $FE$ with apex $c \in C$.
Let's define functor $H$ that sends cone $\lambda: c \Rightarrow F$ to cone $\lambda E: c \Rightarrow FE$ and morphism $f:c \to d$ (between cones $\lambda, \mu$) to "the same" morphism $f:c \to d$ (between $\lambda E ,\mu E$) (it commutes with $\lambda_{j}, \mu_{j}$, hence will also commute with $\lambda_{Ei}, \mu_{Ei}$. So the mapping is correctly defined.) and functor $H'$ that sends cone $\mu: c \Rightarrow F$ to the unique $\lambda$ such that $\lambda E = \mu$ (the unique $\lambda$ exists because $E^{*}$ is fully faithful).
$H, H'$ are inverse to each other, hence the categories of cones are isomorphic (and not just equivalent). I strongly believe I made mistake somewhere when I got this unbelievably strong result, but I have no idea where.
Can someone please tell me where did I go wrong?
Thanks in advance.
EDIT: The thread with old approach is HERE. The exercise is from Emily Riehl's book Category theory in contexts.
There is no mistake: these categories are indeed isomorphic. I guess this statement may be generalized in many ways, but here is an elementary but quite general observation:
Let $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$, $\mathcal{D}$ be categories, $T\colon \mathcal{A}\to\mathcal{C}$, $S\colon \mathcal{B}\to\mathcal{C}$, $R\colon \mathcal{C}\to\mathcal{D}$ be functors, such that $R\circ T$ and $R\circ S$ are injective on objects and $R$ is fully faithful. Then the induced morphism of comma categories $$(\downarrow)_{I_A,R,I_B}\colon(T\downarrow S)\to(R\circ T\downarrow R\circ S)$$ is an isomorphism.
The proof is more or less obvious: if $f\colon T(a)\to S(b)$, $f'\colon T(a')\to S(b')$ and $R(f)=R(f')$, then $a=a'$ and $b=b'$ by the assumptions. Then $(\downarrow)_{I_A,R,I_B}$ is fully faithful (because $I_A$, $I_B$ and $R$ are fully faithful) and bijective on objects, hence it is an isomorphism.
This applies immediately to the categories of cones: take $\mathcal{A}=C$, $\mathcal{B}=\mathbf{1}$, $\mathcal{C}=C^{J}$, $\mathcal{D}=C^I$, $T=\Delta_{J,C}$, $S=\Delta_{F}$, $R=E^*$. It is clear that $R\circ S=\Delta_{FE}$ is injective on objects and $R\circ T=\Delta_{I,C}$ is injective on objects when $I\ne\mathbf{0}$ (case of $I=\mathbf{0}$ is obvious), and $E^*$ is fully faithful as you mentioned. Hence, the induced morphism of comma categories $$(\downarrow)_{I_{C},E^*,I_{\mathbf{1}}}\colon(\Delta_{J,C}\downarrow F)\to(\Delta_{I,C}\downarrow FE)$$ is the desired isomorphism of categories of cones (which coincides with your $H$, actually).