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 $\mathcal{F,G}$)
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.