Let $S_{A \times B}, T_{A \times B} : A \times B \rightarrow C$ and $S_{B \times A}, T_{B \times A} : B \times A \rightarrow C$ be "the same" functors (modulo the order of $A$ and $B$) in the sense that
$$ S_{A \times B} (a,b) = c \text{ iff } S_{B \times A} (b,a) = c \text {, for $a \in ob(A)$, $b \in ob(B)$, and $c \in ob(C)$ } $$
and likewise for $T_{A \times B}$.
If $\eta_{A \times B}$ from $S_{A \times B}$ to $T_{A \times B}$ is natural in $A \times B$ to $C$, does that also mean its corresponding $\eta_{B \times A}$ is also natural from $B \times A$ to $C$?
For any natural transformation $\tau : F \to G$, $\tau_H : F\circ H \to G \circ H$ is also a natural transformation. (Note, this is the horizontal composition $\tau * id_H$.) In your case $S_{B \times A} = S_{A\times B} \circ \sigma$ where $\sigma : B\times A \to A \times B$ is the swapping functor in $\mathbf{Cat}$. $\eta_{B\times A} = \eta_{A\times B,\sigma}$.