The forgetful functor from category of ringed spaces to the category of topological spaces $F\colon RS\to Top$ is a bifibration. The fiber over each topological space $X$ is equivalent to the opposite category ringed sheaves over $X$, hence is complete and cocomplete. Do these facts alone allow us to conclude that the category of ringed spaces is complete and cocomplete?

(Note: I understand the usual construction of co/limits in $RS$ via co/equalizers and co/products.)

My motivation is in proving using this bifibration that the category of locally ringed spaces is complete and cocomplete via the fact that it is a reflective subcategory of $RS$.

Ideally, I would like to know the follow: if a functor $F\colon E\to B$ is a fibration with $B$ and each fiber complete, can we conclude $E$ is complete?


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