# Mono/Epi of sections of presheaves

Let $\mathsf{C}$ be a category with initial and terminal objects, and $\phi:\mathscr{F}\to\mathscr{G}$ a morphism of presheaves on $X$ taking values in $\mathsf{C}$. I have a rather messy proof that if $\phi$ is a monomorphism (resp. epimorphism), then so is $\phi(U):\mathscr{F}(U)\to\mathscr{G}(U)$ for every open subset $U\subset X$. I'm wondering if there is any elegant proof?

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possible duplicate of Monic (epi) natural transformations – Martin Brandenburg Feb 10 '13 at 20:11

Assume $\phi$ is a monomorphism. Assume $\phi(U)\circ f=\phi(U)\circ g$ with $f,g\colon \mathscr G(U)\to A$. Consider presheaf $\mathscr A$ with $\mathscr A(V)=A$ if $U\subseteq V$, $\mathscr A(V)=0$ otherwise (with obvious restrictions). Then $f$ allows us to define $\psi_f\colon\mathscr G\to \mathscr A$ by letting $\psi_f(V)\colon \mathscr G(V)\to \mathscr A(V)$ be the morphism $f\circ \operatorname{res}^V_{U}$ if $U\subseteq V$ (and $0$ otherwise). Similarly, we find $\psi_g$. As $\phi$ is a monomorphism, $\psi_f=\psi_g$, especially, $f=\psi_f(U)=\psi_g(U)=g$, hence $\phi(U)$ is a monomorphism.