202507111607

Tags : Category Theory

Composition of Adjunctions

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Theorem

Given 2 adjunction $F⊣G$ and $F^{\prime }⊣G^{\prime }$ the composites form the adjunction $F{F}^{\prime }⊣G^{\prime }G$.

We can simply show the natural isomorphisms:

$$E(F^{\prime }Fc,e)\cong D(Fc,G^{\prime }e)\cong C(c,G{G}^{\prime }e)$$

Another way would be to define the units and counits as follows:

$$\begin{array}{rl}\bar{\eta }& :=1_C\stackrel{\eta }{\Rightarrow }GF\stackrel{G{\eta }^{\prime }F}{\Rightarrow }G{G}^{\prime }{F}^{\prime }F\\ \overline{ϵ}& :=F^{\prime }FGG\stackrel{F^{\prime }ϵG^{\prime }}{\Rightarrow }{F}^{\prime }{G}^{\prime }\stackrel{{ϵ}^{\prime }}{\Rightarrow }{1}_E\end{array}$$

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References