9.8.8 Functors Corepresentably Full on Cores

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

Definition 9.8.8.1.1 ▶ Functors Corepresentably Full on Cores

A functor $F\colon \mathcal{C}\to \mathcal{D}$ is corepresentably full 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 full.

Remark 9.8.8.1.2 ▶ Unwinding Definition 9.8.8.1.1

In detail, a functor $F\colon \mathcal{C}\to \mathcal{D}$ is corepresentably full on cores if, for each $\mathcal{X}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$ and each natural isomorphism

there exists a natural isomorphism
such that we have an equality
of pasting diagrams in $\mathsf{Cats}_{\mathsf{2}}$, i.e. such that we have

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

Question 9.8.8.1.3 ▶ Characterisation of Functors Corepresentably Full on Cores

Is there a characterisation of functors corepresentably full on cores?

This question also appears as [MO 468125].

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