Sequence of additive functors

Question

I have a problem about this problem:

"Give an example of a sequence of additive functors and maps

$T'\rightarrow T\rightarrow T''$

which is exact on projectives but such that the composition is not zero"

Thank you for your time!

Answer 1

Consider $T' = T = T'' = \text{Tor}_1(-,\mathbb Z/n \mathbb Z):Ab \rightarrow Ab$ with natural transformations $T' \rightarrow T \rightarrow T''$ all being identities.

$\text{Tor}_1(-,\mathbb Z/n \mathbb Z)$ is additive and $\text{Tor}_1(P,\mathbb Z/n \mathbb Z)$ is zero for all projective abelian groups $P$ so the sequence is exact at projectives.

Furthermore $\text{Tor}_1(A,\mathbb Z/n \mathbb Z)$ is not zero for all groups $A$, in general $\text{Tor}_1(A,\mathbb Z/n \mathbb Z) = \{a \in A: na = 0\}$ so the composition $T' \rightarrow T''$ isn't zero.