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How can I prove that additive functors preserve split exact sequences?

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I am assuming we are dealing with a functor $F: R\mathrm{Mod} \to S\mathrm{Mod}$ where $R$ and $S$ are commutative rings, although the result may hold in more general settings that I am not sufficiently familiar with.

A split exact sequence $0 \to A \xrightarrow{i} B \xrightarrow{p} C \to 0$ can be characterized by 4 functions and 5 equations: \begin{align} i &: A \to B, \\ q &: B \to A, \\ j &: C \to B, \\ p &: B \to C, \\ q \circ i &= 1_A, \\ p \circ j &= 1_C, \\ p \circ i &= 0, \\ q \circ j &= 0, \\ i \circ q + j \circ p &= 1_B. \end{align} That is, the given sequence is split exact if and only if there are $j$ and $q$ so that $i, j, p, q$ satisfy the above equations. Now any functor preserves composition and identity, and additive functors also preserve addition and the 0 morphism, so the entire characterization of the split exact sequence is preserved, and hence its image is split exact.