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## EnglishModificar

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### NounModificar

1. Patrono:Category theory 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
1. 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":
1. $F \eta : F \rightarrow FGF$ composed with $\epsilon G: FGF \rightarrow F$ commutes with $\mbox{id}_F: F \rightarrow F$ and
2. $\eta G: G \rightarrow GFG$ composed with $F \epsilon: GFG \rightarrow G$ commutes with $\mbox{id}_G: G \rightarrow G$.[1]
2. 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
1. "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 $\alpha:\mathcal{D}(FX,Y)\rightarrow \mathcal{C}(X,GY)$;
2. "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 $\alpha:\mathcal{D}(FX,Y') \rightarrow \mathcal{C}(X,GY')$.[2]