A functor $F\colon \mathcal{C}\to \mathcal{D}$ is essentially injective if it satisfies the following condition:
- (★) For each $A,B\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, if $F(A)\cong F(B)$, then $A\cong B$.
11.6.5 Essentially Injective Functors
Let $\mathcal{C}$ and $\mathcal{D}$ be categories.
Is there a characterisation of functors $F\colon \mathcal{C}\to \mathcal{D}$ such that:
-
1.
For each $\mathcal{X}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$, the precomposition functor
\[ F^{*}\colon \mathsf{Fun}(\mathcal{D},\mathcal{X})\to \mathsf{Fun}(\mathcal{C},\mathcal{X}) \]is essentially injective, i.e. if $\phi \circ F\cong \psi \circ F$, then $\phi \cong \psi $ for all functors $\phi $ and $\psi $?
-
2.
For each $\mathcal{X}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$, the postcomposition functor
\[ F_{*}\colon \mathsf{Fun}(\mathcal{X},\mathcal{C})\to \mathsf{Fun}(\mathcal{X},\mathcal{D}) \]is essentially injective, i.e. if $F\circ \phi \cong F\circ \psi $, then $\phi \cong \psi $?
This question also appears as [Emily, Characterisations of functors $F$ such that $F^*$ or $F_*$ is [property], e.g. faithful, conservative, etc].
