Let $\mathcal C$ be a category. Why does the Functor $\mathcal F:\emptyset \to \mathcal C$ exist? In general $Ob(\mathcal C)$ is not a set, so $\mathcal F:\emptyset \to Ob(\mathcal C)$ is not a function and i don't have the vacuous condition from the set-theoretic case, right?
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When the set of objects is not a set for you, what is it then? A class perhaps? Note that still for every class $T$ there is a unique map $\emptyset \to T$. Recall that a map $S \to T$ between classes is just a formula $\phi$ (actually a subclass of $S \times T$) such that $\forall s (s \in S \Longrightarrow \exists ! t \in T (\phi(s,t))$. If $S=\emptyset$, this is satisfied for every $\phi$ (but all of them define the same map).