In a category with zero object, it is easy to see that if $0\rightarrow X\overset{f}{\rightarrow} Y$ is exact then $\ker f=0$, since $0\rightarrow X$ is monic and hence is its own image. However, when I try to prove its dual statement, that if $X\overset{f}{\rightarrow}Y\rightarrow0$ is exact then $\operatorname{coker} f=0$, somehow I can't. Am I overlooking certain fact, or is it true at all? Or is it a corroboration of my mounting suspicion that things are not exactly symmetrical between a category and its dual category?
Does $X\overset{f}{\rightarrow} Y\rightarrow0$ being exact imply $\operatorname{coker}f=0$?
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category-theory
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3You've already proved the cokernel thing too! It _does_ follow by duality from the statement you proved for the kernel. It might help to dualize your proof step by step to see what's going on. – 2012-11-30
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0The exactness of $X\overset{f}{\rightarrow} Y\rightarrow0$ implies that $\operatorname{id}_Y:Y\rightarrow Y$ is the image of $f$, so provided that the category has epimorphic images, then it follows that $f$ is an epimorphism. But I can't seem to go anywhere without the assumption. – 2012-11-30