Naturality of a Change of Base Result

Naturality of a Change of Base Result

[C.F.Townsend]

Given a geometric morphism n:E->F between Grothendieck toposes (over, say,
Set) E is both an F-indexed category and a Set-indexed category. There is
the following change of base result for any small (i.e. internal to Set)
category C: -

The category of F-indexed functors p^*C->E is equivalent to the category
of Set-indexed functors C->E (where the first E is as an F-indexed cat and
the second as a Set-indexed cat). (And p:F->Set.)

Is there anything published/known as to the naturality of this equivalence?
It is easy to see that it is natural in functors on C; but I also think (a)
that it may be natural with respect to filtred cocontinuous functors between
inductive completions of C and (b) between filtered cocontinuous functors on
E.

Thanks for any thoughts on this technical question,
Regards, Christopher Townsend (Open University).