A question about (c0)complete categories

Question

Does every complete category have a terminal object, and dually every cocomplete category an initial object?

If so, can someone indicate to me how one can see that?

Answer 1

"Complete" means "has all (small) limits". Terminal objects are limits, specifically limits of the empty diagram. Therefore every complete category has terminal objects by definition. "Cocomplete" is just the dual. Every cocomplete category has initial objects by definition.

Let $\mathbf 0$ be the empty category, i.e. the initial object in the (large) category of locally small categories. By definition, this means there is at most one functor $0_\mathcal C:\mathbf 0 \to \mathcal C$ for any category $\mathcal C$. $\mathcal C$ has a limit of a diagram $D:\mathcal I\to \mathcal C$ for a small category $\mathcal I$ if $\mathsf{Nat}(\Delta(-),D)\cong\mathcal C(-,L)$ for some object $L$ which will be the limit object where $\Delta:\mathcal C\to\mathcal C^\mathcal I$ sends an object to a constant functor and $\mathsf{Nat}(F,G)$ is the set of natural transformations between $F$ and $G$. When $\mathcal I=\mathbf 0$, then there is exactly one functor $\mathcal I\to\mathcal C$ and, if you spell out the definition of natural transformation, you see that there is exactly one natural transformation between that one functor. In other words, $\mathsf{Nat}(\Delta(-),0_\mathcal C)$ is a constantly singleton set functor. Writing $1$ for some arbitrary singleton set, this means that the limit of the empty diagram, i.e. the terminal object, satisfies $1\cong\mathcal C(X, L)$ natural in $X$, but this just says that there is exactly one arrow $X\to L$ for every object $X$. For initial objects, we have $\mathsf{Nat}(0_\mathcal C,\Delta(-))\cong\mathcal C(C,-)$ which is to say $1\cong\mathcal C(C,X)$ natural in $X$ and this means there is exactly one arrow from $C$ to any other object $X$.

Answer 2

https://en.m.wikipedia.org/wiki/Initial_and_terminal_objects

Terminal/ initial are (co)limit on the empty set.