When comparing objects it is often more interesting when two sets are not identical but preserve some form of structure when mapping between them.
relative strength  type and category notation  description  

tightest  Equality  C and D are the same in these terms  
Isomorphism 1_{d} = GF FG = 1_{c} 
There must be an arrow and inverse between the objects and when composed these give the identity map for every element.  
Equivalence 1_{d}GF FG1_{c} 
For equivalence there is a arrow in both directions between the two objects. It is not necessary that GF and FG are the identity elements but they must be isomorphic to the identity elements. 

loosest 

For adjunctions there is a arrow in both directions between the two objects. It is not necessary that GF and FG are the identity elements but only that they have natural transformations to the identity elements. FG is unit (does not change object  injective followed by surjective) but GF does change object (surjective followed by injective). Like equivalence but in one direction. Alternative definition: there is an isomorphism: 
Equivalence Relations
In a less category specific approach to binary relations they may, or may not, have the following properties:
Equivalence Relation  Group  

xy  
transitive 


reflexive  xx  
symmetric 

Comparison of Sets
Here we investigate this comparisons with the simplest concrete categories  sets.
type of 
Sets  GF 

Equality  
Isomorphism 1_{d} = GF FG = 1_{c} 

Equivalence 1_{d}GF FG1_{c} 

Adjunction 


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