Cfrgtkky

I am trying to get the arrow from $R$ to $R[x]$ to cooperate with me and go alongside the LHS of the picture.

Here is my MWE:

begin{tikzcd}

     & R arrow[d, "f"] arrow[ld, "phi"', hook'] arrow[rd, "phi", hook] arrow["i", hook', bend right=135]{dd} &  \

     Rleftlbrackfrac{1}{r}rightrbrack arrow[r, "exists!F", two heads, dashed, hook] & {frac{R[x]}{(rx-1)}} arrow[r, "exists!G", two heads, dashed, hook] & Rleftlbrackfrac{1}{r}rightrbrack \

     & {R[x]} arrow[u, "pi", two heads] arrow[ru, "g"'] & 

end{tikzcd}

I'd also like to know how to get the spiral that indicates that the diagram commutes inside the top left and bottom right triangles. See, for example, the below.

After seeing your figure (sorry did not read carefully enough! it is very dark in our burrow;-).

documentclass{article}

usepackage{tikz-cd,amsmath}

usetikzlibrary{arrows.meta,bending}

begin{document}

begin{tikzcd}[execute at end picture={

    draw[{Hooks[width=+0pt 10.8,length=+0pt 3.6,harpoon,line cap=round]}->] 

    (tikzcdmatrixname-1-2) to[out=175,in=185,looseness=3] 

    node[midway,left]{$i$} (tikzcdmatrixname-3-2);

    draw[-{Latex[bend]},thick] ([xshift=-0.35cm,yshift=0.45cm]tikzcdmatrixname-2-2)

    arc(0:340:0.2);

    draw[-{Latex[bend]},thick] ([xshift=-0.45cm,yshift=-0.45cm]tikzcdmatrixname-2-2)

    arc(20:360:0.2);

    draw[-{Latex[bend]},thick] ([xshift=0.55cm,yshift=-0.45cm]tikzcdmatrixname-2-2)

    arc(20:360:0.2);

    }]

     & R arrow[d, "f"] arrow[ld, "phi"', hook'] arrow[rd, "phi", hook] &  \

     Rleftlbrackfrac{1}{r}rightrbrack arrow[r, "exists!F", two heads, dashed, hook] & {frac{R[x]}{(rx-1)}} arrow[r, "exists!G", two heads, dashed, hook] & Rleftlbrackfrac{1}{r}rightrbrack \

     & {R[x]} arrow[u, "pi", two heads] arrow[ru, "g"'] & 

end{tikzcd}

end{document}