TITLE: Monomorphism into separated presheaf
QUESTION [5 upvotes]: Let $X, Y : \mathbb{C}^\text{op} \to \mathbf{Set}$, where $\mathbb{C}$ is a small category, and assume the natural transformation $\alpha : X \to Y$ is a monomorphism. If $Y$ is separated with respect to a coverage on $\mathbb{C}$, then is $X$ necessarily separated (w.r.t. the same coverage)?

REPLY [3 votes]: Another way of defining sheaves that I find elegant is the following: $Y$ is a sheaf (with respect to a given covering) if and only if for every covering sieve $S$ on $U$ represented as a subfunctor of $\mathsf{Hom}(-,U)$ we have $y\mapsto(f\mapsto Y(f)(y)):Y(U)\to\mathsf{Nat}(S,Y)$ is an isomorphism. Here, $\mathsf{Nat}(S,Y)$ is the set of natural transformations from $S$ to $Y$. $Y$ is a separated presheaf if this is only a monomorphism.
Given that $Y$ is separated and $\alpha$ is a mono, then so is $x\mapsto(f\mapsto Y(f)(\alpha_U(x))):X(U)\to\mathsf{Nat}(S,Y)$ for every $U$ and $S$ covering $U$ because it's a composition of monomorphisms. Naturality of $\alpha$ states that $Y(f)\circ\alpha_U=\alpha_V\circ X(f)$, in other words $$x\mapsto(f\mapsto Y(f)(\alpha_U(x))) = x\mapsto(f\mapsto \alpha(X(f)(x)))$$
This means $x\mapsto(f\mapsto X(f)(x))$ is a mono since it factors through a mono via post-composition by a mono.