Does a cone require its object to be in its diagram?

Question

Basic Category Theory for Computer Scientists says

Let $C$ be a category and $D$ a diagram in $C$. A cone for $D$ is a $C$-object X and arrows $f_i : X \to D_i$ (one for each object $D_i$ in $D$), such that for each arrow $g$ in $D$, the diagram commutes

Does a cone require its object $X$ to be an object in the diagram $D$, or just in $C$?

Thanks.