show that functor $L_0T$ is right exact

Question

Show that $L_0T$ is right exact.Here T is an additive functor from category of $\Lambda$-modules to abelian group.

I try to prove it,but I think the condition is too little.Any hints?

Answer 1

By construction, if you have a short exact sequence $0\to X'\to X\to X''\to 0$, you get a long exact sequence whose initial terms look like $\cdots\to L_1TX'' \to L_0TX'\to L_0TX\to L_0TX''\to 0$ so $L_0T$ is right exact.