Contravariant functor properties

Question

What does F as an exact contravariant additive functor preserves or changes over an abelian category? (i.e kernels, cokernels, images, etc) Thanks

Answer 1

A covariant exact functor ${\bf A}\to{\bf B}$ preserves finite limits and colimits, so a contravariant exact functor does it too as a (covariant) functor on the opposite category: $F:{\bf A}^{op}\to {\bf B}$. Here ${\bf A}^{op}$ is again Abelian (as [at least one form of] the axioms are self dual).

Now limits (in particular, kernels and products) in ${\bf A}^{op}$ are colimits (cokernels, corpoducts) in $\bf A$, and these are mapped to limits in $\bf B$. The image ${\rm im\,}f$ of an arrow $f$ will go to the coimage of $F(f)$, but if $\bf B$ is also Abelian, this coincides with the image ${\rm im\,}F(f)$, so images are preserved.