It is known that the category of schemes is not cocomplete (e.g. see this question: Colimits of schemes). However, do diagrams of schemes for which every morphism is etale have colimits? More generally, I'd really like to know if, given a scheme $X$, if the category of etale schemes over $X$ is cocomplete. I should mention that I know very little of algebraic geometry. I'm interested for "categorical reasons".

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quotientdoesn't exist as an algebraic space, it does not follow that there cannot be a categorical co-equalizer in the category of schemes (which may not be a quotient). Being a quotient is a stronger property. (Admittedly this is a picky point, since if such a co-equalizer exists it would be useless since it lacks other good properties, but nonetheless it is not evident, strictly speaking, if your suggested example is really an example. – Boyarsky Jun 22 '10 at 21:56