Effaceable cohomological delta functors are universal

Set the degree-zero component to be the given map $u^0$.

Suppose by induction that for some $n\ge 0$ we have already constructed natural transformations $u^i:T^i\Rightarrow S^i$ for all $i\le n$, compatible with the connecting morphisms through degree $n-1$. For each object $A$, choose an injective effacement $e_A:A\rightarrowtail I_A$ for $T^{n+1}(A)$ using [L2]. The cohomological clause of [L3], which is valid for every $n\ge 0$, defines a map $u_A^{n+1}:T^{n+1}(A)\to S^{n+1}(A)$, and [L3] makes it independent of the chosen effacement.

Naturality and compatibility with the connecting morphisms follow exactly as in the homological case: use the covered naturality from [L3] on a common dominating injective effacement and then appeal to the choice-independence from [L3]. Thus adjoining $u^{n+1}$ extends the partial morphism one degree further.

Uniqueness in degree $n+1$ comes from the cokernel description in [L2]: once the degree-$n$ component is fixed, item 19 gives only one possible map out of $T^{n+1}(A)$. Hence the inductive extension is unique in every degree, and [L1] shows that $T$ is universal.