Krdytkyu

I'm currently working on the following exercise from Emily Riehl's Category Theory in Context,

Exercise 3.1.xii. Suppose $E:I stackrel{simeq}{to} J$ defines an equivalence between small categories and consider a diagram $F : J to C$. Show that the category of $J$-shaped cones over $F$ is equivalent to the category of $I$-shaped cones over $FE$, and use this equivalence to describe the relationship between limits of $F$ and limits of $FE$.

Here $J$ is assumed to be small and $C$ locally small, although I am not sure this assumptions are needed. I have found a similar idea to mine here, which is to define the following functor

$$
begin{align}
Gamma : int &mathbf{Cone}(_,F) longrightarrow intmathbf{Cone}(_,FE) \
& (c,(lambda_j)_{jin J}) longmapsto (c,(lambda_{Ei})_{iin I}) \
(f:c to d &text{ s.t. }lambda_jf = mu_j) longmapsto (f:c to d text{ s.t. } lambda_{Ei}f = mu_{Ei})
end{align}
$$

that takes $lambda : c Rightarrow F$ to the natural transformation that has its components for each object $Ei$, and takes a morphism that commutes with the cones to itself (since it will still commute with the selected legs). Now, it is asserted in the linked post that $Gamma$ should be an isomorphism of categories (in particular an equivalence), but the details aren't specified and I haven't managed to finish the job. So far, here is what I have come up with:

$Gamma$ is (essentially) surjective: take a cone $nu : c Rightarrow FE$. For each $j in J$, since $E$ is essentially surjective take $varphi_j : Ei_j xrightarrow{sim} j$ be an isomorphism. Now if we define $mu_j$ to be the composite
$$
c xrightarrow{nu_{i_j}} FEi_j xrightarrow{Fvarphi_j} Fj
$$
then $mu$ is a cone over $c$: if $f: j to j'$ is an arrow on $J$, then $varphi_{j'}^{-1}fvarphi_j: Ei_j to Ei_{j'}$ must come from a (unique) arrow $s : i_j to i_{j'}$ such that $Es = varphi_{j'}^{-1}fvarphi_j$. Consequently,
$$
begin{align}
Ffmu_j &= FfFvarphi_jnu_{i_j} = F(fvarphi_j)nu_{i_j} = Fvarphi_{i_{j'}}(FEs)nu_{i_j} \ & =Fvarphi_{i_{j'}}(FEs)nu_{i_j} = Fvarphi_{i_{j'}}nu_{i_{j'}} = mu_{j'}
end{align}
$$
and so in effect $mu$ is a cone over $c$. Since we can in particular take each $varphi_{Ei}$ to be $1_{Ei}$ for each $j$ such that $j = Ei$, we get that $Gamma(c,mu) = (c,nu)$.

It suffices to see now that $Gamma$ is fully faithful (to prove the equivalence, at least). It is clear to me that if $f,g: c to d$ are distinct morphisms that commute with the respective cones over $c$ and $d$, by construction we get that $Gamma f = f neq g = Gamma g$.

However, if we have $f: c to d$ such that it commutes with the legs of the cones that correspond to the image of $E$, why should it be that $f$ commutes with all legs of the original cones?

Let $f:cto d$ be such that it commutes with the cones in the image of $E$; and let $jin J$.

Let $g:jto Ei$ be an isomorphism. You have the following commutative diagrams (sorry for the formatting, I don't know how to do triangles on here)

$require{AMScd} begin{CD}
c @>{id_c}>> c\ @V{lambda_j}VV @VV{lambda_{Ei}}V\
Fj @>>{Fg}> FE_i
end{CD}$ because $(c,lambda)$ is a cone

$require{AMScd} begin{CD}
d @>{id_d}>> d\ @V{mu_{Ei}}VV @VV{mu_j}V\
FEi @>>{Fg^{-1}}> Fj
end{CD}$ because $(d,mu)$ is a cone

$require{AMScd} begin{CD}
c @>{f}>> d\ @V{lambda_{Ei}}VV @VV{mu_{Ei}}V\
FEi @>>{id_{FEi}}> FE_i
end{CD}$ by hypothesis

Then you can put these three diagrams side by side in the order (1-3-2) to get

$require{AMScd} begin{CD}
c @>{f}>> d\ @V{lambda_j}VV @VV{mu_j}V\
Fj @>>{id_{Fj}}> Fj
end{CD}$ because $id_dcirc fcirc id_c = f, Fg^{-1}circ id_{FE_i}circ Fg = id_{Fj}$

and this is exactly what you want. So this comes down to the facts that $(c,lambda)$, $(d,mu)$ are cones over all $J$, that $F$ is a functor and that $E$ is essentially surjective.