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Aluffi's "Chapter 0", on pg. 492, says the following:

Let $C$ and $D$ be categories, and let $\mathcal{F}:C\to D, \mathcal{G}:D\to C$ be functors. We say that $\mathcal{F}$ and $\mathcal{G}$ are adjoint, if there are natural isomorphisms $$Hom_C(X,\mathcal{G}(Y))\overset{\sim}\longrightarrow Hom_D(\mathcal{F}(X),Y)$$

I interpreted the $\overset{\sim}\longrightarrow$ to mean a bijection. However, I've been told that I should pay attention to the phrase "natural isomorphism". What does a natural isomorphism mean in this context? I used to think that a natural isomorphism can only be between two functors mapping the same category (say category C) to the same category (say category D). Here we have two functors with different domains and ranges (the functors $\mathcal{F,G}$)

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  • $\begingroup$ For one thing, the Homs are sets, so the natural isomorphisms would be bijections of sets that are "natural". If/when the Hom-sets have more structure, such as abelian groups, those admit comparison, etc. $\endgroup$ – paul garrett Jun 24 '16 at 21:44
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What you actually have are two functors $C^{op} \times D \to \mathsf{Set}$, namely

$${\rm Hom}_C(-, \mathcal{G}(-))$$

and

$${\rm Hom}_D(\mathcal{F}(-), -)$$

And you want a natural isomorphism between them.

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