Contents

category theory

# Contents

## Definition

A full and faithful functor is a functor which is both full and faithful. That is, a functor $F\colon C \to D$ from a category $C$ to a category $D$ is called full and faithful if for each pair of objects $x, y \in C$, the function

$F\colon C(x, y) \to D(F(x), F(y))$

between hom sets is bijective. “Full and faithful” is sometimes shortened to “fully faithful” or “ff.” See also full subcategory.

###### Remark

It is not sufficient for there simply to exist some isomorphism between $C(x, y)$ and $D(F(x), F(y))$. For instance, consider the category comprising a parallel pair $f, g : x \rightrightarrows y$ and the identity-on-objects endofunctor $F$ sending $f \mapsto f$ and $g \mapsto f$. We have $C(x, y) \cong C(F(x), F(y)) = C(x, y)$, but this functor is not fully faithful.

## Properties

• Fully faithful functors are closed under pushouts in Cat. For ordinary categories this was proven by Fritch and Latch; for enriched categories it is proven in Stanculescu, Prop. 3.1, and for (∞,1)-categories it is proven in Simspon, Cor. 16.6.2.

• Fully faithful functors $F : C \to D$ can be characterized as those functors for which the following square is a pullback, where the vertical maps are source and target, and the horizontal maps are induced by $F$

$\array{ C^{} &\to& D^{} \\ \downarrow && \downarrow \\ C \times C &\to& D \times D }$
• The bijections exhibiting full faithfulness of $F$ form a natural isomorphism, by functoriality of $F$ and of pre- and postcomposition.

• Let $I L \dashv R$ be an adjunction. If $I$ is fully faithful, then $L \dashv R I$. In this case, the two adjunctions induce the same monad. This is Proposition 1.1 of DFH75. (For a converse, see dominant functor.)