Coproduct of the initial object with itself

Question

This is a really basic question, and I apologize, but I really want to make sure I'm not doing something very stupid.

Let $\mathcal{C}$ be a category, $x\in\mathcal{C}$ a fixed object. Consider the under category $x\backslash\mathcal{C}$ of arrows $x\to y$ in $\mathcal{C}$ and commutative triangles as morphisms. Let $0 := \left(x\xrightarrow{1_x}x\right)\in x\backslash\mathcal{C}$. Is it true that the coproduct $0\sqcup0$ is isomorphic to $0$?

In fact, if $0$ is the (an) initial element (as is the case above), is it true that $0\sqcup0\cong0$?

Answer 1

In any category, the coproduct of an object $A$ with an initial object $0$ is simply $A$. Just check the universal property with respect to the inclusion morphisms $0 \to A$ (given by the definition of $0$) and $\mathit{id}:A \to A$.

Answer 2

This coproduct will be initial regardless of the category. Think about the co-cones associated with morphisms from such a coproduct to any object $A$; how many pairs of morphisms $(f:0\to A, g:0\to A)$ are there?