CatDat

Implication Details

Claim: Given a functor whose domain has equalizers, if it is conservative and preserves preserves equalizers, then it is faithful.

Proof: Let $f,g : X \rightrightarrows Y$ be two morphisms in the domain, and choose an equalizer $E \hookrightarrow X$. By assumption, $F(E) \to F(X)$ is the equalizer of $F(f),F(g) : F(X) \rightrightarrows F(Y)$. Thus, if $F(f) = F(g)$, then $F(E) \to F(X)$ is an isomorphism. Since $F$ is conservative, $E \to X$ is an isomorphism, which means $f = g$.