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I have some difficulty understanding the following proof (source):

Claim: If $T: \mathcal{A} \rightarrow \mathcal{B}$ is a right exact functor of two Abelian categories, then $L_0 T$ and $T$ are canonically naturally equivalent.

Proof: Take a projective resolution $P_\bullet$ of some object $A\in \mathcal{A}$. Then we have an exact sequence $P_1 \rightarrow P_0 \rightarrow A \rightarrow 0$. As $T$ is assumed right exact, we have an exact sequence $TP_1 \rightarrow TP_0 \rightarrow TA \rightarrow 0$. Hence $H_0(TP_\bullet)\simeq TA$.

My difficulty is seeing this final conclusion. I could prove that this was indeed true for the more specific case of R-Modules, but there I used the first isomorphism theorem, which I'm not sure if exists in an arbitrary Abelian category. Could you please explain carefully how the above equivalence follows from all of this?

Edit A category C is abelian if it is 1) additive, 2) for every monomorphism $f: A \rightarrow B$ the pair $(A,f)$ is the kernel of the epimorphism $B \rightarrow coker f$, 3) For every epimorphism $f: A\rightarrow B$ the pair $(f,B)$ is a cokernel of the monomorphism $ker f \rightarrow A$.

gen
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  • Of course we have the first isomorphism theorem, it's part of the definition of abelian category ! – Maxime Ramzi May 21 '19 at 16:02
  • @Max, hm I didn't know about that. Could you point me to some expository materials that spell this out in detail? – gen May 21 '19 at 16:04
  • Well from what wikipedia tells me, different people have different definitions; but see here for instance : http://www.math.ucla.edu/~ggim/S14-212lecturenote.pdf Anyway, the point is that "$A\to B\to C\to 0$ is exact" implies that "$\mathrm{coker}(A\to B)\to C$ is an isomorphism" – Maxime Ramzi May 21 '19 at 16:09
  • @EricTowers: C is abelian if 1) additive, 2) for any epi $A \rightarrow B$ the pair $(B,f)$ is a cokernel of the morphism $ker(f) \rightarrow A$, 3) for any mono $f: A \rightarrow B$ the pair $(A,f)$ is the kernel of the morphism $B \rightarrow coker(f)$. – gen May 21 '19 at 16:13
  • @EricTowers just added – gen May 22 '19 at 08:16

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