I can see two ways to read this proposition (it's probably the language barrier problem -- I'm not an English native speaker), and I suppose the first way below is what Leinster intended to convey. Is that right?
There's a one-to-one correspondence between [limit cones on D] and [representations of $Cone(-,D)$]. Furthermore, any representing object of $Cone(-,D)$ is a limit object of $D$.
There's a one-to-one correspondence between [limit cones on D] and [those representations of $Cone(-,D)$ for which the representing object is a limit object of $D$].
From what I understand, a priori, there are more [representations of $Cone(-,D)$] than [those representations of $Cone(-,D)$ for which the representing object is a limit object of $D$]. And I think my proof (which I'm not giving here) shows that there's a bijection as described in 1 (the fact that the representing object is the limit is not needed as a hypothesis).