For a counterexample, consider $R = \mathbb{C}$. The automorphism $i \colon \mathbb{C} \to \mathbb{C}$ given by complex conjugation is certainly a ring homomorphism, but it isn't $\mathbb{C}$-linear.

For a counterexample, consider $R = \mathbb{C}$. The automorphism $i \colon \mathbb{C} \to \mathbb{C}$ given by complex conjugation is certainly a ring homomorphism, but it isn't $\mathbb{C}$-linear.

Edit: Another example would be the slice $p / \text{Set}$, where $p$ is a one element set. This category is the category of pointed sets, and not all maps between two sets will preserve the chosen point.