I've already asked here about the following confusing restatement of the definition of limit preservation from Leinster's book (p. 137):
And now a new question has arisen. As far as I understand from the answer given in the above link, this equivalent definition from the screenshot can be reformulated as follows:
A functor $F: \mathscr A\to\mathscr B$ preserves limits if the following property is satisfied: whenever $D:I\to\mathscr A$ is a diagram that has a limit $(\lim D,p_i:\lim D\to D(i))_{i\in I}$, the composite $F\circ D: I\to\mathscr B$ also has a limit $(\lim(F\circ D), q_i: F\circ D\to FD(i))_{i\in I}$, and the unique arrow $\alpha: F(\lim D)\to\lim (F\circ D)$ with the property $p_i=q_i\circ \alpha$ for all $i\in I$ (whose existence is guaranteed by the definition of the limit of $F\circ D$) is an isomorphism.
Is this indeed equivalent to what Leinster says? (I tried to get rid of the notion of components of the arrow). The reason I'm in doubt is that at the bottom of the screenshot there's a remark saying that both sides of (5.22) should be isomorphic in a specific way. But I don't see how my definition is talking about a specific isomorphism, it's just saying that there is some isomorphism.
If my definition isn't correct, how to fix it (avoiding the use of the notion of components)? If my definition is correct, where does my definition contain that "particular isomorphism" Leinster is talking about?