9.8.4 Functors Representably Faithful on Cores

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

Definition 9.8.4.1.1 ▶ Functors Representably Faithful on Cores

A functor $F\colon \mathcal{C}\to \mathcal{D}$ is representably faithful on cores if, for each $X\in \text{Obj}\webleft (\mathsf{Cats}\webright )$, 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 9.8.4.1.2 ▶ Unwinding Definition 9.8.4.1.1

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

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

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

then $\alpha =\beta $.

Question 9.8.4.1.3 ▶ Characterisation of Functors Representably Faithful on Cores

Is there a characterisation of functors representably faithful on cores?

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