11.8.7 Functors Corepresentably Faithful on Cores

Let $\mathcal{C}$ and $\mathcal{D}$ be categories.

Definition 11.8.7.1.1 ▶ Functors Corepresentably Faithful on Cores

A functor $F\colon \mathcal{C}\to \mathcal{D}$ is corepresentably faithful on cores if, for each $X\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$, the postcomposition by $F$ functor

\[ F_{*}\colon \mathsf{Core}\webleft (\mathsf{Fun}\webleft (\mathcal{X},\mathcal{C}\webright )\webright )\to \mathsf{Core}\webleft (\mathsf{Fun}\webleft (\mathcal{X},\mathcal{D}\webright )\webright ) \]

is faithful.

Remark 11.8.7.1.2 ▶ Unwinding Definition 11.8.7.1.1

In detail, a functor $F\colon \mathcal{C}\to \mathcal{D}$ is corepresentably faithful on cores if, given a diagram of the form

if $\alpha $ and $\beta $ are natural isomorphisms and we have

\[ \alpha \mathbin {\star }\operatorname {\mathrm{id}}_{F}=\beta \mathbin {\star }\operatorname {\mathrm{id}}_{F}, \]

then $\alpha =\beta $.

Question 11.8.7.1.3 ▶ Characterisation of Functors Corepresentably Faithful on Cores

Is there a characterisation of functors corepresentably faithful on cores?

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