I am not very familiar with the analytic side of the pictures or with the analytification functor but here is what I can claim, it seems from your comment that this answer your question:

If you have two subcanonical site $C$ and $D$ (it means that representable presheaves are sheaves, so it is the case with your example).

If $C$ has all finite limits.

If you have a functor $f :C \rightarrow D$ that preserves finite limits and send covering family in $C$ to covering family in $D$ then:

the precompostion with $f : Prsh(D) \rightarrow Prsh(C)$ send sheaves over $D$ to sheaves over $C$ and the restriction of this functor to sheaves defines the $f_*$ part of a geometric morphism:

$f_*: sh(D) \rightarrow sh(C)$

Moreover (easy to check from the universal property) $f^*$ send representable sheaves in $sh(C)$ to representable sheaves in $sh(D)$ in a way that extend $f: C \rightarrow D$.

You have more general claim of this sort on the nLab page [morphism of sites][1]

Or in more standard litterature, have a look to MacLane&Moerdijk "sheaves in geometry and logic" chapter VII, more specifically setion 9 and 10. [1]: https://ncatlab.org/nlab/show/morphism+of+sites

I am not very familiar with the analytic side of the pictures or with the analytification functor but here is what I can claim, it seems from your comment that this answer your question:

If you have two subcanonical site $C$ and $D$ (it means that representable presheaves are sheaves, so it is the case with your example).

If $C$ has all finite limits.

If you have a functor $f :C \rightarrow D$ that preserves finite limits and send covering family in $C$ to covering family in $D$ then:

the precompostion with $f : Prsh(D) \rightarrow Prsh(C)$ send sheaves over $D$ to sheaves over $C$ and the restriction of this functor to sheaves defines the $f_*$ part of a geometric morphism:

$f_*: sh(D) \rightarrow sh(C)$

Moreover (easy to check from the universal property) $f^*$ send representable sheaves in $sh(C)$ to representable sheaves in $sh(D)$ in a way that extend $f: C \rightarrow D$.

You have more general claim of this sort on the nLab page morphism of sites

Or in more standard litterature, have a look to MacLane&Moerdijk "sheaves in geometry and logic" chapter VII, more specifically setion 9 and 10.