Adjunction

==English==
===Noun===
# A form of similarity between a pair of categories \mathcal{C} and \mathcal{D} which is weaker than equivalence, which in turn is weaker than isomorphism. Given functors F:\mathcal{C} \rightarrow \mathcal{D} and G:\mathcal{D} \rightarrow \mathcal{C} , F is "left adjoint" of G, and G "right adjoint" of F, denoted as F \dashv G , if 
## there are a pair of natural transformations \eta: \mbox{id}_\mathcal{C} \rightarrow GF and \epsilon: FG \rightarrow \mbox{id}_\mathcal{D} satisfying the following "triangle identities": 
### F \eta : F \rightarrow FGF composed with \epsilon G: FGF \rightarrow F commutes with \mbox{id}_F: F \rightarrow F and
### \eta G: G \rightarrow GFG composed with F \epsilon: GFG \rightarrow G commutes with \mbox{id}_G: G \rightarrow G .TheCatsters – Adjunctions 1
## there is a natural isomorphism \alpha: \mathcal{D}(FX,Y) \cong \mathcal{C}(X,GY) , which is natural in the sense of being "natural in X and Y", where
### "natural in X" means that for every f:X\rightarrow X', \alpha:\mathcal{D}(FX',Y) \rightarrow \mathcal{C}(X',GY) composed with \mathcal{C}(f,GY):\mathcal{C}(X',GY) \rightarrow \mathcal{C}(X,GY) commutes with \mathcal{D}(Ff,Y):\mathcal{D}(FX',Y) \rightarrow \mathcal{D}(FX,Y) composed with \alpha:\mathcal{D}(FX,Y)\rightarrow \mathcal{C}(X,GY);
### "natural in Y" means that for every f:Y \rightarrow Y', \alpha:\mathcal{D}(FX,Y) \rightarrow \mathcal{C}(X,GY) composed with \mathcal{C}(X,Gf):\mathcal{C}(X,GY) \rightarrow \mathcal{C}(X,GY') commutes with \mathcal{D}(FX,f):\mathcal{D}(FX,Y) \rightarrow \mathcal{D}(FX,Y') composed with \alpha:\mathcal{D}(FX,Y') \rightarrow \mathcal{C}(X,GY').TheCatsters – Adjunctions 2
===References===
====Translations====
* Bulgarian: съединение
pl:adjunction
ru:adjunction
tr:adjunction
vi:adjunction