Corollary 4.3.10 in Leinster's text says that $H_A\cong H_{A'}\iff A\cong A'\iff H^A\cong H^{A'}$.
He writes: "the corollary tells us that two objects are the same if and only if they look the same from all viewpoints (Figure 4.2)". But from what I understand, "looking the same from all viewpoints" only means that $H_A(B)\cong H_{A'}(B)$ for all $B$, and it doesn't subsume naturality. Is this correct?
On a similar note, he writes "For a general category, Corollary 4.3.10 tells us that two objects are the same if they have the same generalized elements of all shapes." But again, having the same generalized elements of all shapes doesn't subsume naturality, from what I can tell...
Then he gives this example:
In each of these three sub-examples he doesn't assume that $H_A(B)\cong H_{A'}(B)$ for all $B$, nor does he assume naturality. So are the conclusions of these sub-examples not related to the corollary? For example, does the conclusion "$A$ and $A'$ have isomorphic underlying sets" in the second sub-example come from a separate "lemma" about the existence of a bijection between $H_A(\mathbb Z)$ and $A$?
And similarly, in the example below, I suppose he is not using Corollary 4.3.10; he seems to be using just our knowledge about sets, right?
