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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?

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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.

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    $\begingroup$ I.e. if $i$ is the initial object of $D$, $t$ is the terminal object of $D$ then constant functor $\Delta_i$ that maps all objects from $C$ into $i$, is the initial object of $[C,D]$ and $\Delta_t$ is the terminal one? $\endgroup$
    – Ivan
    Dec 10 '18 at 12:43
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    $\begingroup$ exactly! a lot of such stuff happens due to your natural transofrmation jsut happening in the target and since all your maps have to be unique (by assumtion on $t$ and $i$) this is again initial and terminal. $\endgroup$
    – Enkidu
    Dec 10 '18 at 12:44

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