The relationship is based on convenience. Antony explains to Octavius that Lepidus is "a slight, unmeritable man" of few ideas, and what ideas he has are second-hand and out-of-date. He will carry loads, and then, when he has served his purpose, be put out to pasture - rather like Antony's horse.
Then, in Act V, Antony's authority is challenged by the young Octavius, when he confronts Antony over who will command the right flank of the army. Octavius calmly asserts his will: "I do not cross you; but I will." (V.i.20)
Compare this to the relationship between Brutus and Cassius, the two leaders of the conspiracy. How do these two men feel about each other? Is political or expediency stronger than affection?