4

I need to shift this diagonal arrow (in bottom box) down.

\xymatrix@R.8pc@C.8pc{
  \bullet \ar[rrr]^{i'^{!}} \ar[ddd]_{Th(\Omega_{f'})} &&& \bullet \ar[ddd]^{Th(\Omega_{f})} \\
  &&& \\
  & \ar@{<=}[ur] && \\
 \bullet \ar[rrr]^{i'^{!}} \ar[ddd]_{f'^{*}} &&& \bullet \ar[ddd]^{f^{*}}\\ 
  & \ar@{<=}[ur]_{E_{x}^{!,*}} && \\ \\
\bullet \ar[rrr]_{i^{!}} &&& \bullet
}
3

What's the problem with writing it like in the first box? I mean like this:

\xymatrix@R.8pc@C.8pc{
  \bullet \ar[rrr]^{i'^{!}} \ar[ddd]_{Th(\Omega_{f'})} &&& \bullet \ar[ddd]^{Th(\Omega_{f})} \\
  &&& \\
  & \ar@{<=}[ur] && \\
 \bullet \ar[rrr]^{i'^{!}} \ar[ddd]_{f'^{*}} &&& \bullet \ar[ddd]^{f^{*}}\\ 
  &&& \\
  & \ar@{<=}[ur]_{E_{x}^{!,*}} &&\\
\bullet \ar[rrr]_{i^{!}} &&& \bullet
}

enter image description here

You can also try a bend arrow:

\xymatrix@R3pc@C3pc{
  \bullet \ar[r]^{i'^{!}} \ar[d]_{Th(\Omega_{f'})} & \bullet \ar[d]^{Th(\Omega_{f})} \\
 \bullet \ar[r]^{i'^{!}} \ar[d]_{f'^{*}} \ar@/^/@{<=}[ur] & \bullet \ar[d]^{f^{*}}\\ 
\bullet \ar[r]_{i^{!}} \ar@/^/@{<=}[ur]_{E_{x}^{!,*}} & \bullet
}

enter image description here

Also next time please post a fully-compilable MWE.

1

One can shift an arrow up or down by

\ar@<5pt>[r] \ar@<-5pt>[r]

One can also shift labels, the vertices themselves

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