The outgoing photon attaches to the electron line before the incoming photon, so the virtual electron carries momentum $k = p - q'$.
The computational chain
From diagram to number—every step of the amplitude calculation.
Crossed Compton diagram
$$i\mathcal{M}_u = (-ie)^2\,\bar{u}(p')\gamma^\mu\frac{i(\not{p}-\not{q}'+m)}{(p-q')^2-m^2}\gamma^\nu u(p)\,\epsilon_\mu(q)\epsilon_\nu^*(q')$$
The photon vertices appear in the opposite order compared to the s-channel: the outgoing photon attaches before the incoming one. The virtual electron carries momentum $p - q'$ with $k^2 = u$. Both channels must be summed for gauge invariance.