Rrerku

In making commutative diagrams in tikz-cd I've been wanting to add in 'mapping' arrows that describe what particular elements map to, parallel to the main diagram arrows.

I've been doing it in the following way, which seems fairly clumsy since the 'map' arrows are separated from the main diagram at a large distance, when ideally they would be close by and parallel to the corresponding maps. What would be a better way to do this?

documentclassarticle
usepackagetikz-cd

begindocument
begintikzcd
a arrow[r, maps to] & phi(a) & \
Rarrow[r, hook, "phi"] arrow[dr, hook, "iota"] & K & phi(k) \
a arrow[dr, maps to] & Q(R) arrow[u, dotted, "exists ! Phi"'] & fraca1 arrow[u, maps to] \
 & fraca1 & 
endtikzcd
enddocument

This is my proposal

documentclass[tikz]standalone
usetikzlibraryarrows,cd,positioning
tikzset %https://tex.stackexchange.com/a/252544/156344
begindocument
begintikzpicture
node (r) at (0,0) $R$;
node (k) at (3,0) $K$;
node (q) at (3,-3) $Q(R)$;
node (ra) at (0,.8) $a$;
node (ka) at (3,.8) $phi(a)$;
node (kb) at (4,0) $phi(k)$;
node (qb) at (4,-3) $frac a1$;
path (q) node[below left=1em and 1em] (qc) $frac a1$;
path (r) node[below left=1em and 1em] (rc) $a$;
draw[right hook->] (r)--(k) node[midway,above] $scriptstylephi$;
draw[right hook->] (r)--(q) node[midway,above right] $scriptstyleiota$;
draw[dotted,->] (q)--(k) node[midway,right] $scriptstyleexists!Phi$;
draw[|->] (ra) edge (ka) (qb) edge (kb) (rc) edge (qc);
endtikzpicture
enddocument

You can reduce the spacing, but in my opinion it's clearer and more elegant if you make two diagrams.

documentclassarticle
usepackageamsmath
usepackagetikz-cd

begindocument

[
begintikzcd
a arrow[r, maps to] & phi(a) &[-2em] \[-4ex]
Rarrow[r, hook, "phi"] arrow[dr, hook, "iota"] & K & phi(a) \
& Q(R) arrow[u, dotted, "exists ! Phi"'] & fraca1 arrow[u, maps to] \[-10ex]
a arrow[dr, maps to] \
& fraca1 & 
endtikzcd
]

[
begintikzcd
Rarrow[r, hook, "phi"] arrow[dr, hook, "iota"] & K &
a arrow[r, maps to,"phi"] arrow[dr, maps to,"iota"] & phi(a) \
& Q(R) arrow[u, dotted, "exists ! Phi"'] &
& fraca1 arrow[u, maps to,"Phi"']
endtikzcd
]

enddocument