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Take the following definition of adjunction from the nlab

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This definition can be found in numerous other places. My brain parses this definition perfectly up till the point where it says that for every $c$ and $d$ the hom-sets are naturally isomorphic. At his point it does SCRREEETCH, because, as a beginner, learning category theory, natural isomorphisms were only defined between functors as a collection of morphisms in the target category and not *between collections of morphisms lying in different categories.

I know that there are other equivalent definitions of adjunctions, but I don't have the time to go through them to see how to make sense of this definition. I just want a clean explanation how to understand this definition.

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    $\begingroup$ $\text{Hom}_D(L(c),d)$ and $\text{Hom}_C(c,R(d))$ are both sets, so they're objects of the category $\text{Set}$, which is the target category where the natural isomorphism is taking place. It's not that the Hom sets are naturally isomorphic - there's just a bijection between them, and these bijections cohere to a natural ismorphism between the Hom functors. $\endgroup$ – Alex Kruckman Mar 14 '16 at 22:16
  • $\begingroup$ The second sentence is just repeating what the first sentence is saying in more basic terms. If you unfold the definition of natural isomorphism you will see this. $\endgroup$ – Zhen Lin Mar 14 '16 at 22:17
  • $\begingroup$ Why does the nlab page say "natural isomorphism" for what has traditionally been called a "natural equivalence"? $\endgroup$ – Rob Arthan Mar 14 '16 at 22:25
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    $\begingroup$ @Rob Arthan natural isomorphism is very standard, to my mind moreso than natural equivalence. $\endgroup$ – Kevin Arlin Mar 14 '16 at 23:13
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    $\begingroup$ @KevinCarlson: is it? I use Mac Lane's "Categories for the Working Mathematician" as a primary source for basic category theory notions. If "natural isomorphism" has taken over from "natural equivalence", then do you call "natural transformations" "natural homomorphisms"? $\endgroup$ – Rob Arthan Mar 14 '16 at 23:23
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Converting my comment to an answer, as suggested:

$\text{Hom}_D(L(c),d)$ and $\text{Hom}_C(c,R(d))$ are both sets, i.e. they're objects of the category $\mathsf{Set}$, which is the target category where the natural isomorphism is taking place.

You wrote:

My brain parses this definition perfectly up till the point where it says that for every c and d the hom-sets are naturally isomorphic.

It's not that the $\text{Hom}$ sets are naturally isomorphic - there's just a bijection between them, and these bijections cohere to a natural ismorphism between the $\text{Hom}$ functors.

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  • $\begingroup$ Thanks! If you are interested, please see also this question, which I raised as a follow-up to your answer, as one point in it still was unclear to me: math.stackexchange.com/questions/1698375/… $\endgroup$ – user10324 Mar 15 '16 at 9:48

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