#
nLab

strong adjoint functor

### Context

#### Category theory

**category theory**

## Concepts

## Universal constructions

## Theorems

## Extensions

## Applications

# Contents

## Definition

For $C$ a cartesian closed category and $L,R : C \to C$ two endofunctors, they are called **strong adjoints** to each other if there is a natural isomorphism

$[L X, A] \simeq [X, R A]$

for all objects $X,A \in C$ and for $[-,-]$ the internal hom.

## Properties

Notice that for $*$ the terminal object of $C$ we have that the global points of the internal hom give the external hom set

$\Gamma [X,A] := C(*, [X, A]) \simeq C(X,A)
\,.$

Therefore strongly adjoint functors are in particular adjoint functors in the ordinary sense.

## References

For instance appendix 6 of

Created on December 7, 2011 19:45:19
by

Urs Schreiber
(131.174.40.86)