Symbiolgy usefully generalizes this. Mating partners are only one of many types of partner in symbiotic relationships.
Sometimes, mutualists pick their partners, predators pick their prey and parasites pick their victims.
These processes are well-characterized as being forms of partner selection.
Paralleling "female choice" there are also "predator choice" and "parasite choice". Mates are selected - but also, so are hosts, prey, food, trading partners, assistants, targets of mimicry, bodyguards, slaves and tennents.
Sexual selection was famously involved in introducing minds into the evolutionary process - making a lie out of the idea that natural selection is "blind". However sexual selection rather unfairly gets all the attention - while other forms of partner selection are neglected - apparently through not having such a distinguished founder and such a catchy name.
Predation and parasitism likely existed before sexual recombination - giving these forms of partner selection a more primitive status. Their toll in modern times is impressive: far more organisms are lost to predation and parasitism than are neglected in the competition for mates. The term "prey selection" has seen some use. However, an umbrella term is also clearly needed. Evolutionists should make more use of the idea of "partner selection".