Existence of Initial Object implies Existence of Terminal Object?

Question

Consider an initial object always exists in a category. Reversing all arrows now. Does this guarantee the existence of terminal objects in that category?

Answer 1

To help you understand what is going on, consider the finite category $\mathcal I$ consisting of three objects $X, Y, Z$ and two non-identity arrows $f\colon Y\to X$ and $g\colon Y\to Z$. Then the initial object of $\mathcal I$ is $Y$. Then $Y$ is the terminal object in the opposite category (ie reversing all arrows) $\mathcal I^{op}$ since $f^{op}\colon X\to Y$ and $g^{op}\colon Z\to Y$.

So, yes, an initial object becomes the terminal object in the opposite category. This is the same as saying that initial and terminal are dual concepts.

I think user18921 said "No" because he took your "that category" to mean the original non-opposite category that you started with. But as we can see by looking at $\mathcal I$, it does not have a terminal object even though it has an initial object $Y$. The dual to this is that $\mathcal I^{op}$ does not have an initial object even though it has a terminal object $Y$.