Swallowtail butterfly black swallowtail is the state butterfly of Oklahoma. Swallowtail butterfly Swallowtail butterflies are large, colorful butterflies in the family Papilionidae, and include over 550 species. Though the majority are tropical, members of the family inhabit every continent except Antarctica. The family includes the largest butterflies in the world, the birdwing butterflies of the genus "Ornithoptera". Swallowtails have a number of distinctive features; for example, the papilionid caterpillar bears a repugnatorial organ called the osmeterium on its prothorax. The osmeterium normally remains hidden, but when threatened, the larva turns it outward through a transverse dorsal groove by inflating it with fluid.
Comma category exists. Lawvere showed that the functors formula_169 and formula_170 are adjoint if and only if the comma categories formula_171 and formula_172, with formula_173 and formula_174 the identity functors on formula_175 and formula_3 respectively, are isomorphic, and equivalent elements in the comma category can be projected onto the same element of formula_177. This allows adjunctions to be described without involving sets, and was in fact the original motivation for introducing comma categories. If the domains of formula_178 are equal, then the diagram which defines morphisms in formula_179 with formula_180 is identical to the diagram which defines a natural transformation formula_181. The