I am a little confused by the last two lines of Weibel's proof. Note that Weibel defines a $\delta$-functor $W_*$ to be universal when incoming morphisms $T_* \to W_*$ are completely characterized by $T_0 \Rightarrow W_0$.
The $\delta$-functor we wish to prove is universal is the left derived $\delta$-functor of a right exact functor $F$ from a category with enough projectives. Weibel defines natural transformations $\phi_n: T_n \Rightarrow L_n F$ and shows they are unique. Then he gets to the last bit where he needs to show that, given a SES, the relevant ladder commutes.
I understand why the squares at the end commute but not why $T(g) \circ \delta$ and $LF(g) \circ \delta$ are the desired maps.