Let $C$ and $D$ are 2 non empty categories and $[C,D]$ is the category of functors between $C$ and $D$. I assume that the constant functor $\Delta_t$ (that maps all objects from $C$ into a single object $t$ from $D$) is the terminal object in the category. I also think that it always exists. Am I right?
Another question is about initial object in the category. What it is?
Best Answer
It depends on the constant functor! since all your natural transformations actually happen on the image category, you need that the image if your object is the terminal respectively final object. Hence, the terminal object is the constant functor onto a $\underline{\textrm{terminal}}$ object, and dually the initial object is the constant functor onto the $\underline{\textrm{initial}}$ object (assuming that $C$ admits those)! which is something you need, otherwise you run into problems fast.