Two functors $F: C \mapsto D$ and $G: D \mapsto C$ are said to be adjoint if there exists a natural bijection
$\tau_{A,B}: Mor(F(A), B) \mapsto Mor(A, G(B)) \quad \forall A \in C \; \forall B\in D.$
I am trying to make sense of the word "natural" here. My understanding is that formally, we view $\tau$ as a natural transformation between two functors
$$Mor(F(-), \star) , Mor(-, G(\star)): C^{\text{op}} \times D \mapsto \text{SET},$$
in which case naturality just means commutativity of the usual diagrams. But, in some references, I found that naturality means that for each (fixed) $A \in C$, $\tau_{A,B}$ is natural in $B$, and for each (fixed) $B \in D$, $\tau_{A,B}$ is natural in $A$. Are these two definitions equivalent? If yes, what is the best way to see that?