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Consider the category of $R$-modules. I am trying to see how i can express a short exact sequence in terms of kernels and cockerels, and how this description can be used to prove that a conservative functor (i.e. a functor that reflects isomorphisms) that preserves limits will also reflect exact sequences.

In particular, we know that a conservative functor that preserves limits, also reflects limits. Additionally, kernels and cockerels are particular cases of equalizers, which can be expressed as limits and colimits.

So suppose $M \stackrel{f}{\rightarrow} N \stackrel{g}{\rightarrow} P$ is a sequence (not necessarily a complex) and suppose that its image under a contravariant conservative functor (lets say from the category of $R$-modules to the category of abelian groups) $F(P) \stackrel{F(g)}{\rightarrow} F(N) \stackrel{F(f)}{\rightarrow} F(M)$ is exact. The exactness can be captured by the relation $Coker(F(g)) = \frac{F(N)}{Ker(F(f))}$. How can we show that this relation is "reflected" to the relation $Coker(f) = \frac{N}{Kerg}$?

My efforts: without using limits/colimits, i showed that $g \circ f=0$ i.e. $Im(f) \subseteq Ker(g)$. However, it seems harder to show the reverse inclusion.

Note: This question i a follow-up on my previous question Injective Cogenerators in the Category of Modules over a Noetherian Ring of which i understand the given answer (but not how to implement the hints of the comments).

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