Let $Sh(\mathsf{\mathbb{C}-fAlg}^{op})$ be the topos of zariski sheaves on finitely genertaed $\mathbb{C}$-algebras.

A complex analytic space for our purpose is a locally ringed space locally isomorphic to an analytic subset of $\mathbb{C}^{n}$ (with the sheaf of holomorphic functions). Denote the category of these by $\mathsf{An}$ (for analytic). There's a grothendiek topology on the category given by jointly sujective open immersions (where open immersion are topological embeddings with locally isomorphic structure sheaves). Let $Sh(An)$ be the topos of analytic sheaves on analytic spaces. We have an affine analytification functor:

$$f: \mathsf{\mathbb{C}-fAlg}^{op} \to \mathsf{An}$$

Which associates to an affine $\mathbb{C}$ scheme the corresponding analytic subspace of $\mathbb{C}^n$. This morphism induces a pushforward for presheaves:

$$f_*: Psh(\mathsf{An}) \to Psh(\mathsf{\mathbb{C}-fAlg}^{op})$$

I'm afraid to say anything about sheaves at this point since I'm not so comfortable with toposes yet really. What I'd like to be able say is the following:

(

Extremely conjectural!) The pushforward $f_*$ extends to a morphism of topoi:$$F_*: Sh(\mathsf{An}) \to Sh(\mathsf{\mathbb{C}-fAlg}^{op})$$

This morphism is geometric with left adjoint the analytification functor:

$$F^*: Sh(\mathsf{\mathbb{C}-fAlg}^{op}) \to Sh(\mathsf{An})$$

In partuicular this functor sends schemes in $Sh(\mathsf{\mathbb{C}-fAlg}^{op})$ to analytic spaces.

**Is there a way to make this precise?**

I'm trying to see the analytification functor come out of general nonsense because it seems like the existence of it should be natural.

I'm obviously not talking about any powerful statements about properties of this functor like GAGA etc. I'm only looking for a natural way to define this functor.