11.8.4 Functors Representably Faithful on Cores

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

Definition 11.8.4.1.1 ▶ Functors Representably Faithful on Cores

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

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

is faithful.

Remark 11.8.4.1.2 ▶ Unwinding Definition 11.8.4.1.1

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

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

then $\alpha =\beta $.

Question 11.8.4.1.3 ▶ Characterisation of Functors Representably Faithful on Cores

Is there a characterisation of functors representably faithful on cores?

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