The incoming photon is absorbed by the electron, producing a virtual electron with momentum $k = p + q$, which then emits the outgoing photon.
The computational chain
From diagram to number—every step of the amplitude calculation.
The diagram
The incoming photon is absorbed at vertex 1, creating a virtual electron with momentum $k = p + q$ and $k^2 = s$. The virtual electron then emits the outgoing photon at vertex 2.
Feynman rules expression
$$i\mathcal{M}_s = (-ie)^2\,\epsilon_\mu(q)\,\epsilon_\nu^*(q')\,\bar{u}(p')\gamma^\nu\frac{i(\not{p}+\not{q}+m)}{(p+q)^2-m^2}\gamma^\mu u(p)$$
Unlike Moller or Bhabha, this is a single fermion line: the electron enters, absorbs a photon, propagates, emits a photon, and exits. The two $\gamma$ matrices are sandwiched between the same pair of spinors. The external photons contribute polarization vectors $\epsilon_\mu$ and $\epsilon_\nu^*$.