Implication Details

Assumptions: conservative, equalizer-preserving

Assumptions on source category: equalizers

Conclusions: faithful

Reason: Let $f,g : X \rightrightarrows Y$ be two morphisms in the source category, 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 $f = g$ .