= Pointwise monomorphism in a set-valued functor category
{title2=$\alpha\text{ monic}\iff\forall A,\ \alpha_A\text{ injective}$}
= Pointwise monic natural transformation
{synonym}
A <natural transformation> between <functors> to the <Category of sets> is a <monomorphism> in the <functor category> exactly when every component is an <injective function>. Componentwise cancellation proves sufficiency. For necessity, the <Yoneda lemma> converts two elements with equal component images into two transformations from a covariant <representable functor>; monicity cancels these transformations, forcing the elements to coincide. The argument remains valid for a <locally small category> in an ambient universe where the functor category is formed.
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