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We have the left and right adjoint of the forgetful functor from ${\bf Top}$ to ${\bf Set}$. The right adjoint equips any set with the indiscrete topology and the left adjoint equips any set with the discrete topology.

So, what are the unit and counit to these adjoints?

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    Do you have any thoughts? – 2017-01-27
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    No, these concepts are kind of new to me. So, I'm stuck. – 2017-01-27
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    Well, what is the definition of the unit map of an adjuntion? – 2017-01-27
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    Given an adjunction, (let $C$ to $D$ be the categories and forgetful function go from $C$ to $D$; so right adjoint, R, goes from $C$ to $D$ and left adjoint, L, goes from $D$ to $C$) there is a natural transformation from $1_{C}$ to $RL$ called the unit. The counit would be a natural transformation from $LR$ to $1_{D}$ – 2017-01-27
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    So, for the right adjoint R from $C=$Top to $D = $Set and left adjoint L from $D = $Set to $C=$Top, the unit would be the natural transformation from $1_{Top}$ to $RL$. But I don't know what else to say about this. – 2017-01-27
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    The adjunction is a **specific** natural bijection between certain hom-sets, hom(FX,Y)=hom(X,GY). *Knowing* the adjuction (and you wrote that you know those two adjuntions…) means knowing that bijection. Now take, for example, X equal to GY: on the right hand side you have the identity map of GY, and on the left there is a corresponding morphism $FGY\to Y$: that is the counit. – 2017-01-27
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    If you know the adjunction, then you know its unit and counit. – 2017-01-27
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    I guess I'm confused about what exactly a adjoint is. Like I can read the definition and described it above. I guess I don't understand how the adjoint's I described above are indeed functors and that the resulting isomorphsim is natural in both variables. – 2017-01-27
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    Well, can you then tell me what is your definition for what it means for two functors to be mutually adjoint? – 2017-01-27

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