There is no trivial way to start with a list of initial-state and final-state particles and determine which channels are involved. You just have to start enumerating diagrams, and working out if they are allowed or not.
To do that you use a small set of tools
- Each fundamental interaction has an allowed set of diagrams. You have to know or look these up.
- Each vertex must conserve four-momentum.
- Each exernal line has to be on-shell (have the correct mass) but internal lines can be off-shell.
Then you start drawing and discard any diagram that fails to meet one of those rquirements.
In general it is difficult to enumerate all possible diagrams, but you have a two-in-two-out reaction and have specified "tree level", so there are only three topologies and only one internal line (where you get a choice of identity) in each of them. You've also specified "QED" so I'm going to limit myself to electromagnetic and weak interactions (strict QED doesn't inlude the weak interactions, but I did weak physics for most of my career, so I include it by default).
You just go to town.
Take the s-channel mentioned in the comments for an example. On the incident side the only diagrams with two charged lepton legs have either a photon or a $Z^0$ on the third leg, and that must be the particle that participates in the vertex with the two out-going photons. Can either of those bosons do that?
Doing the t- and u-channels is left as an exercise, but I'll give you a hint: the internal line will be off-shell or you won't be able to conserve four-momentum at the vertices.