Can someone explain why the forelimbs of a cat, a bat, and a whale would be homologous structures?
There are usually two types of criteria used to determine homology: similarity and congruence. The first one takes into account overall similarity (same bones in the same position, same nerves, blood vessels, muscles, etc), ontogeny (similar stages of development from the same origins), and body position relative to other elements. These similarities outweigh the functional differences given by the adaptation of the same limb to different functions (walking, flying, swimming respectively, in this case). The second criterium is congruence: similarities thought to be homologous are tested against other traits. Cats, bats and whales are placental mammals, and they all share many other characters that suggest their close relationships; thus the hypothesis that their forelimbs are homologous structures agrees with their being mammals, amniotes, tetrapods, lobe-finned fish, and gnathostomates (paired limbs first appeared in the latter).