Lack of uniqueness for unit/counit of adjunctions? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T09:15:38Z http://mathoverflow.net/feeds/question/98209 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/98209/lack-of-uniqueness-for-unit-counit-of-adjunctions Lack of uniqueness for unit/counit of adjunctions? Eitan Chatav 2012-05-28T18:31:47Z 2012-05-28T18:31:47Z <p>Hi, assuming we have $\mathbb{K}$-linear categories and functors $F:D\to C$ and $G:C\to D$ which are adjoint $F\dashv G$, then there exist unit and counit functorial morphisms $\varepsilon:FG\to 1_C$ and $\eta:1_D\to GF$ such that</p> <p>$(\varepsilon F)(F\eta)=1_F$ and $(G\varepsilon)(\eta G)=1_G$.</p> <p>Am I right that the unit and counit are not unique but that any non-zero scalar multiples $c\varepsilon$ and $\frac{1}{c}\eta$ are also unit/counits?</p>