Let $\mathcal{C}$ and $\mathcal{D}$ be categories.
A functor $F\colon \mathcal{C}\to \mathcal{D}$ is conservative if it satisfies the following condition:1
Let $F\colon \mathcal{C}\to \mathcal{D}$ be a functor.
Characterisations. The following conditions are equivalent:
The functor $F$ is conservative.
For each $f\in \operatorname {\mathrm{Mor}}(\mathcal{C})$, the morphism $F(f)$ is an isomorphism in $\mathcal{D}$ iff $f$ is an isomorphism in $\mathcal{C}$.
Interaction With Fully Faithfulness. Every fully faithful functor is conservative.
Interaction With Precomposition. The following conditions are equivalent:
For each $\mathcal{X}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$, the precomposition functor
is conservative.
The equivalent conditions of Item 5 of Proposition 11.6.1.1.2 are satisfied.
Similarly, $F(\operatorname {\mathrm{id}}_{A})=F(f^{-1}\circ f)$. But since $F$ is fully faithful, we must have
showing $f$ to be an isomorphism. Thus $F$ is conservative.
Is there a characterisation of functors $F\colon \mathcal{C}\to \mathcal{D}$ satisfying the following condition:
This question also appears as [Emily, Characterisations of functors $F$ such that $F^*$ or $F_*$ is [property], e.g. faithful, conservative, etc].
