Counterexample. [math-000T]

The result is $H$ can be not natural.

This counterexample is given by Zhixuan Yang:

Let $\mathcal{C}$ be the two-object one-arrow category, and $\mathcal{D}$ be $Set$ , and

where $2$ is the two elements set. For $0$ we choice $H(0) = id$ and $H(1) = \text{not}$ , then $H$ is not natural.