Given a category $\mathbb{C}$ and a graph $\mathbb{G}$, we have the category of diagrams where its objects are called diagrams, which are graph morphisms $D:\mathbb{G} \to \mathbb{C}$.
A morphism of the category of diagrams, denoted $\tau: D \to E$, is a family $(\tau_x:D(x) \to E(x))_{x \in \mathbb{G}_0}$, where $\mathbb{G}_0$ is the set of objects for $\mathbb{G}$, such that, for every morphism $u:x \to y$ in $\mathbb{G}$, $\tau_y{D(u)} = E(u)\tau_x$.
My question is that since $D(x)$ and $E(x)$ are objects in $\mathbb{C}$, $\tau_x:D(x) \to E(x)$ has to be a morphism in $\mathbb{C}$. But how do we know for sure that such a morphism exists in $\mathbb{C}$?
Any help will be greatly appreciated.
If such morphism does not exist then also no morphism $D\to E$ exists in the category of diagrams.
We are dealing with necessary (not per se sufficient) conditions here.