Let $\mathcal{C,D}$ be two categories and $F:\mathcal{C}\to\mathcal{D}$, $G:\mathcal{D}\to\mathcal{C}$ be two functors. We say that $(F,G)$ is a pair of adjoint functors when $$\text{Hom}_\mathcal{D}(F-,-)\simeq\text{Hom}_\mathcal{C}(-,G-), $$ that is, these functors are naturally isomorphic. Ι have some trouble understanding this and I cannot find a precise meaning to it.
Does that mean that whenever we fix an object $X\in\text{Ob}\mathcal{C}$ the functors $\text{Hom}_\mathcal{D}(FX,-)$ and $\text{Hom}_\mathcal{C}(X, G-)$ are naturally isomorphic and whenever we fix an object $Y\in\text{Ob}\mathcal{D}$ the functors $\text{Hom}_\mathcal{D}(F-,Y)$ and $\text{Hom}_\mathcal{C}(-,GY) $ are naturally isomorphic?
The most direct way to understand this isomorphism is by viewing these functors as having domain $\mathcal{C}^{op}\times\mathcal{D}$. In particular, the left side is the composite $$\mathrm{Hom}_\mathcal{D}\circ F^{op}\times id_\mathcal{D}:\mathcal{C}^{op}\times\mathcal{D}\to\mathcal{D}^{op}\times\mathcal{D}\to\mathbf{Set}$$ and the right is $$\mathrm{Hom}_\mathcal{C}\circ id_{\mathcal{C}^{op}}\times G:\mathcal{C}^{op}\times\mathcal{D}\to\mathcal{C}^{op}\times\mathcal{C}\to\mathbf{Set}$$
This turns out, in this particular case, to be equivalent to what you've guessed above with respect to fixing one of the arguments. The benefit of the above presentation is you can describe it as a single ordinary natural isomorphism instead of a family of natural isomorphisms between families of functors.