A nonnatural choice of connecting maps does not form a delta functor

Start with the homology delta functor from [L2]. Keep every functor $H_n$ unchanged and keep every connecting map unchanged except on one chosen short exact sequence with nonzero connector from [L3], where replace $\mathrm{\partial }$ by $-\mathrm{\partial }$. Each individual long exact sequence remains exact.

Choose a distinct isomorphic copy of the cone sequence in [L3], and alter the connector on only the original sequence. The chosen isomorphism of short exact sequences induces isomorphisms on the two homology groups. Before the alteration, naturality identifies the two routes around the connecting square with the same map $\pm 1:\mathbb{Z}\to \mathbb{Z}$. After changing exactly one connector to its negative, the two routes are opposite maps $\pm 1$ and $\mp 1$, which are unequal over $\mathbb{Z}$. Hence this naturality square fails, and [L1] shows that the altered family is not a homological delta functor.