Given a field $K$ and a vector space $V$ of dimension at least $1$ we can define the projective space $PV=(V - \{0\})/K^\times$ to be the quotient of $V$ by the action of the multiplicative group of $K$. This gives us a functor $P$ on the category of all finite-dimensional vector spaces of dimension at least $1$ with the arrows being injective linear maps.
Are there any issues that arise if we define $P(0)=\emptyset$? It seems reasonable since $\mathrm{PGL}(V)=\mathrm{GL}(V)/\mathrm{ZGL}(V)$ and hence $\mathrm{PGL}(0) = 1=\mathrm{Aut}(\emptyset)$.
If this works, then why not extend this idea to all Grassmannians: define $\mathrm{Gr}_k(V)$ as usual if $\mathrm{dim}(V) \geq k$, and otherwise $\mathrm{Gr}_k(V) = \emptyset$. This would make the Grassmannians functors on the wide subcategory of vector spaces with injective maps.
This seems to work as far as I can tell, but maybe I'm missing something obvious. I apologize if this is something standard in algebraic geometry; I'm a differential geometer and I don't have much experience with algebraic geometry.