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$\newcommand{\fs}{\mathscr{F}}\newcommand{\gs}{\mathscr{G}}$Let
${\bf C}$ be some algebraic category (e.g. commutative rings),
${\bf Psh}(X,{\bf C})$ the category of presheaves on $X$ taking
values in ${\bf C}$, and ${\bf Sh}(X,{\bf C})$ the category of
sheaves on $X$ taking values in ${\bf C}$. Let $\phi:\fs\to\gs$
be an epimorphism in ${\bf Sh}(X,{\bf C})$. Is it possible that
$\phi$ is not an epimorphism in ${\bf Psh}(X,{\bf C})$ ?  (Sheaf
monomorphism seems to imply presheaf monomorphism, if one
assumes that every presheaf can be sheafified. The proof doesn't
extend to epimorphisms.)  share|improve this question At least
in the category of (pre)sheaves of sets, epimorphisms of
presheaves are the naive ones, i.e., they are surjective on
sections over every object. Epimorphisms of sheaves can be
characterized as the morphisms which induce surjections on all
stalks. It's definitely possible to have an epimorphism of
sheaves which is not an epimorphism of the underlying
presheaves, a