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".