An incoming photon produces a virtual electron, which absorbs the second photon and produces the outgoing electron-positron pair. Related to pair annihilation by time reversal and to Compton scattering by crossing.
The computational chain
From diagram to number—every step of the amplitude calculation.
Diagram topology
$$\gamma(q_1) + \gamma(q_2) \;\longrightarrow\; e^-(p') + e^+(k')$$
Photon $q_1$ converts into a virtual electron with momentum $k = q_1 - p'$ at vertex 1, producing the outgoing electron $p'$. The virtual electron (with $k^2 = t$) then absorbs photon $q_2$ at vertex 2, producing the outgoing positron $k'$. This is a single fermion line connecting the two external fermions, with both photons attached along it — the same topology as Compton scattering and pair annihilation, related by crossing.
Feynman rules expression
$$i\mathcal{M}_t = (-ie)^2\,\epsilon_\mu(q_1)\,\epsilon_\nu(q_2)\,\bar{u}(p')\gamma^\mu\frac{i(\not{q}_1-\not{p}'+m)}{(q_1-p')^2-m^2}\gamma^\nu v(k')$$
Both external photons are incoming, so both carry polarization vectors $\epsilon_\mu(q_1)$ and $\epsilon_\nu(q_2)$ (no complex conjugation). The outgoing electron gives $\bar{u}(p')$ on the left, and the outgoing positron gives $v(k')$ on the right. The virtual electron propagator carries the numerator $\not{k}+m$ with $k = q_1 - p'$, and the denominator is $(q_1 - p')^2 - m^2 = t - m^2$, where $t = (q_1 - p')^2$.
Crossing relation to pair annihilation
$$\overline{|\mathcal{M}(\gamma\gamma \to e^+e^-)|^2} = \overline{|\mathcal{M}(e^+e^- \to \gamma\gamma)|^2}$$
Pair production is the time-reversed (crossed) version of pair annihilation $e^+e^- \to \gamma\gamma$. By CPT invariance, the spin-averaged squared amplitudes are equal when expressed in terms of the same Mandelstam invariants $s$, $t$, $u$. The functional form of $\overline{|\mathcal{M}|^2}$ is identical for both processes; the cross sections differ only through the initial-state flux factor and the final-state phase space.
Spin and polarization sum — trace structure
$$\overline{|\mathcal{M}_t|^2} = \frac{1}{4}\frac{e^4}{(t-m^2)^2}\,\mathrm{Tr}\!\left[(\not{p}'+m)\gamma^\mu(\not{q}_1-\not{p}'+m)\gamma^\nu(\not{k}'-m)\gamma^{\nu'}(\not{q}_1-\not{p}'+m)\gamma^{\mu'}\right]\cdot(-g_{\mu\mu'})(-g_{\nu\nu'})$$
Average over the two incoming photon polarizations ($\frac{1}{2}\cdot\frac{1}{2}$) and sum over outgoing electron and positron spins. The photon polarization sums each give $\sum_\lambda \epsilon_\mu^\lambda\epsilon_\nu^{\lambda*}\to -g_{\mu\nu}$ (valid in Feynman gauge). The electron completeness relation gives $\sum u\bar{u} = \not{p}'+m$, and the positron completeness gives $\sum v\bar{v} = \not{k}'-m$ (note the minus sign). The propagator numerator $(\not{q}_1-\not{p}'+m)$ appears twice — once from $\mathcal{M}$ and once from $\mathcal{M}^*$.
Result equals pair annihilation
$$\overline{|\mathcal{M}_t|^2} \supset -2e^4\,\frac{m^2 - s}{m^2 - t}$$
After contracting the metric tensors and evaluating the trace using $\gamma^\mu\gamma_\alpha\gamma_\mu = -2\gamma_\alpha$ and standard trace identities, the result has the same functional form as the pair-annihilation t-channel formula. This is guaranteed by CPT: the squared amplitude is a Lorentz-invariant function of $s$, $t$, and $u$, and the two processes share the same values of these invariants. The cross section differs from annihilation only through the initial-state flux factor and the final-state phase space.