# 3.Preserves and Reflects

## 11.06.2020

### Contents/Index

1. Definition
2. Full, Faithful and Embeddings
@3. Preserves and Reflects
4. Contra-, Co- and Bivariant
5. Co- and Contravariant Hom Functors

Given two categories $\mathcal{C},\mathcal{D}$, and a functor $F : \mathcal{C} \rightarrow \mathcal{D}$, we say that $F$

• preserves a propery $P$ if whenever an object or arrow, $c$, of $\mathcal{C}$ has $P$, then so does $F(c)$.
• reflects a property $P$ if whenever an object or arrow, $d = F(c)$, of $\mathcal{D}$ has $P$, then so does $c$.