the definition of invertible sheaf on a functorial scheme in category theory We define a functorial scheme as in "Two functorial definitions of schemes".A invertible sheaf is important and I'm interested in category theory, so I hope to define invertible sheaf in category theory like a scheme. However, the definition of a invertible sheaf(or a locally free sheaf) require a ringed space. Therefore to define a invertible sheaf for a functorial scheme seems to be difficult.
My question is: Can we define invertible sheaf on a scheme in category theory?
Thanks in advance.
 A: I presume you already know how to define a sheaf of groups/rings/modules on a scheme-defined-as-a-sheaf.  If not then you will have to start there. There are basically only two more ingredients needed to define invertible sheaves:

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*The structure sheaf of a scheme is represented by the scheme $O = \operatorname{Spec} \mathbb{Z} [x]$. This is a ring object in the category of schemes, so for every scheme $X$, the set of morphisms $X \to O$ has a natural ring structure. This defines a sheaf of rings on the category of all schemes, but precomposing it with the forgetful functor gives you a sheaf $O_X$ on the category of schemes over $X$ (or on the small Zariski site of $X$ – take your pick).


*An invertible sheaf $M$ on $X$ is a sheaf of $O_X$-modules that is locally isomorphic to $O_X$. Locally isomorphic means there is a cover of $X$ consisting of morphisms $U \to X$ such that pulling back $M$ along the morphism yields an $O_U$-module isomorphic to $O_U$. (Strictly speaking this is the definition of a locally free sheaf of rank 1... but as you know, they are the same thing as invertible sheaves.)
Notice that the above makes equal sense for schemes-defined-as-sheaves and schemes-defined-as-ringed-spaces.
