What is the image of the empty category under a given functor?

Question

I'm pretty sure this is obvious, but just making sure. Does the empty category always map (via functor) to the empty subcategory of the codomain (category)?

Answer 1

Your question is very poorly worded. It sounds like you're asking what the image of the functor $F:\varnothing\to C$ is, where $C$ is any category and $\varnothing$ is the empty category, with no objects and no morphisms. Is this correct?

If so, then there is exactly one such functor $F$ for every category $C$ (in other words, $\varnothing$ is the initial category). Its image is the empty category, regarded as a subcategory of $C$.