Given a category $\mathbb{C}$ and its object $C$, the contravariant hom functor based at $C$ is a contravariant functor $hom_{\mathbb{C}}(-,C):\mathbb{C} \to Sets$, where $Sets$ is the category of sets, carrying a morphism $f:A \to B$ in $\mathbb{C}$ to the map $hom_\mathbb{C}(f,C):hom_\mathbb{C}(B,C) \to hom_\mathbb{C}(A,C)$ defined $\alpha \mapsto \alpha{f}$.
As we can use an alternative definition for any contravariant functor using a covariant functor from $\mathbb{C}^{op}$, my notes say as follows.
Given a category $\mathbb{C}$ and its object $C$, the contravariant hom functor based at $C$ is a covariant functor $hom_{\mathbb{C}}(-,C):\mathbb{C}^{op} \to Sets$, where $Sets$ is the category of sets, carrying a morphism $f:B \to A$ in $\mathbb{C}^{op}$ to the map $hom_\mathbb{C}(f,C):hom_\mathbb{C}(A,C) \to hom_\mathbb{C}(B,C)$ defined $\alpha \mapsto \alpha{f}$ (in $\mathbb{C}$).
I strongly believe this is a typo and it should read as follows.
Given a category $\mathbb{C}$ and its object $C$, the contravariant hom functor based at $C$ is a covariant functor $hom_{\mathbb{C}}(-,C):\mathbb{C}^{op} \to Sets$, where $Sets$ is the category of sets, carrying a morphism $f:B \to A$ in $\mathbb{C}^{op}$ to the map $hom_\mathbb{C}(f,C):hom_\mathbb{C}(B,C) \to hom_\mathbb{C}(A,C)$ defined $\alpha \mapsto \alpha{f}$ (in $\mathbb{C}$).
Any help will be greatly appreciated.