No, this is not the case.

Let the category C be vector spaces (say over the real numbers). Given any real number we get a natural transformation of the identity functor on C. On components for a given vector space V, this transformation is defined to be multiplication by the given real number.

If we whisker this (on the target side) with the product functor we get an infinite family of natural transformations from $\times$ to $\times$.

You can get a more exotic example by noting that a pair of real numbers gives an automorphism of $C \times C$, hence by wiskering (on the source side) we get another family. In components this is the transformation which on $V \oplus W$ scales V by the first number and W by the second.

No, this is not the case.

Let the category C be vector spaces (say over the real numbers). Given any real number we get a natural transformation of the identity functor on C. On components for a given vector space V, this transformation is defined to be multiplication by the given real number.

If we whisker this (on the target side) with the product functor we get an infinite family of natural transformations from $\times$ to $\times$.

You can get a more exotic example by noting that a pair of real numbers gives an automorphism of the identity functor of $C \times C$, hence by wiskering (on the source side) we get another family. In components this is the transformation which on $V \oplus W$ scales V by the first number and W by the second.