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Cokernels are "dual" to kernels. I've been told that if X dual to Y and there's a theorem about X then there's a "dual" theorem about Y.
Theorem: In $\boldsymbol{Grp}$ a homomorphism has trivial kernel iff it is injective.
However it is not true that a group homomorphism has trivial cokernel iff it's surjective. So what would instead be the dual theorem to the above theorem? (or have I been lied to and those dual theorems don't always exist?)
The dual theorem holds in the opposite category $\boldsymbol{Grp}^{op}$, which is not equivalent to $\boldsymbol{Grp}$.
This sort of duality holds in general when a theorem is true for all categories, or at least some subset of categories that are self dual. If for all categories $\mathcal C$, some theorem holds for $\mathcal C$, then the dual theorem holds for $\mathcal C^{op}$. Since the original theorem also holds for the category $\mathcal C^{op}$, the dual theorem holds for $\mathcal (C^{op})^{op} = \mathcal C$.
This particular theorem doesn't hold for all categories (where kernels, cokernels, injectivity and surjectivity can be defined). Thus, the dual theorem is not guaranteed to hold.
