CatDat

Implication Details

Claim: Given a functor whose domain is balanced, if it is faithful, then it is conservative.

Proof: It is easy to see that a faithful functor $F$ reflects monomorphisms: If we have two morphisms $x_1, x_2 : U \rightrightarrows X$ and $f : X \to Y$ such that $f(x_1) = f(x_2)$, and $F(f)$ is a monomorphism, then $F(x_1) = F(x_2)$; therefore, $x_1 = x_2$, so $f$ is also a monomorphism. The dual argument shows that $F$ also reflects epimorphisms. Therefore, if $F(f)$ is an isomorphism, then $f$ is both a monomorphism and an epimorphism; by the assumption on the domain category, this implies that $f$ is an isomorphism.