Item 2 (017H) — The Clowder Project

Table of Contents
Part 4: Categories
Chapter 11: Categories
Section 11.8: Even More Conditions on Functors
Subsection 11.8.9: Functors Corepresentably Fully Faithful on Cores
Item 2 (cite)

11.8.9 Functors Corepresentably Fully Faithful on Cores

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

Definition 11.8.9.1.1 ▶ Functors Corepresentably Fully Faithful on Cores

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

Remark 11.8.9.1.2 ▶ Unwinding Definition 11.8.9.1.1

In detail, a functor $F\colon \mathcal{C}\to \mathcal{D}$ is corepresentably fully faithful on cores if it satisfies the conditions in Remark 11.8.7.1.2 and Remark 11.8.8.1.2, i.e.:

1.
For all diagrams 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 $.

2.
For each $\mathcal{X}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$ 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

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