Consider a cohomological delta functor $T^*:\mathscr{A}\to\mathscr{B}$ between abelian categories such that $T^i$ is an effaceable functor for all $i>0$, i.e. for all $i>0$, and for any $A\in\mathscr{A}$, there exists an injection $u_A:A\hookrightarrow M_A$ in $\mathscr{A}$ such that $T^i(u_A) = 0$. Grothendieck famously proved in his Tōhoku paper that this implies that $T^*$ is a universal $\delta^*$-functor, which he later on used to prove his Vanishing Theorem.

However, in his Tōhoku paper, Grothendieck introduces a tremendous amount of unique terminology and definitions, a bit different from other authors of homological algebra (e.g. Weibel), so I've been having a bit of trouble following both the proof itself, and his work up to that proof, so I was wondering if there is an easier proof.

First, let's fix some $\delta^*$-functor $S:\mathscr{A}\to\mathscr{B}$, and suppose we are given $f^0:T^0\to S^0$. My general idea is to go by induction (as I see that Grothendieck does), and assume that for some $n\ge 1$, we have defined $f^i:T^i\to S^i$ for all $0\le i<n$ commuting with all connecting morphisms. Then it is sufficient to show that we can uniquely define $f^n:T^n\to S^n$ commuting with the connecting morphisms, i.e. for any exact sequence $$0\to A\to B\to C\to 0$$ in $\mathscr{A}$, the following square commutes: $$ \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex} \newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}} \begin{array}{c} T^{n-1}(C) & \ra{\delta^n} & T^n(A)\\ \da{f^{n-1}_C} & & \da{f^n_A}\\ S^{n-1}(C) & \ras{\delta^n} & S^n(A) \\ \end{array} $$

Now, my first thoughts are to fix the maps $u_A$, and use them to define a short exact sequence $$0\to A\xrightarrow{u_A}M_A\xrightarrow{p_A}P_A\to 0$$for all $A\in\mathscr{A}$, then if we let this is exact sequence induce the connecting maps ${^T\!\!\delta_A^n}:T^{n-1}(C)\to T^n(A)$ and ${^S\!\!\delta_A^n}:S^{n-1}(C)\to S^n(A)$, however, all this gives me is that ${^T\!\!\delta_A^n}$ is surjective. Is this enough to define $f_A^n$?