|Greater noctule bat|
At least in the parts of the world with a temperate climate, the great majority of bat species are insectivorous. Of course, there are a number of vegetarian bats, especially in the tropics, including the large fruit bats of the Old World as well as smaller, often fig-eating, species in the Americas. But insect-eating does seem the default, even though there is general agreement these days that bats are more closely related to the larger carnivores and to the hoofed mammals than they are to the other small insectivores, such as shrews and hedgehogs.
From an ecological, trophic web, perspective, insectivores are a type of carnivore - after all, they eat other animals, even if they're small ones. But, when talking about mammals (and similarly sized vertebrates) most researchers tend to draw a distinction between those that hunt invertebrate prey and those that eat comparatively large vertebrates. So there's a distinction between leopards and killer whales on the one hand, and shrews and anteaters on the other.