In Hakim’s book “Topos annelés et schémas relatifs”, Chap. III, Def. 2.3 states that a ringed topos $(X,A)$ is a locally ringed topos when two equivalent conditions are satisfied:

(i) For each $U in X$ and each section $s in A(U)$ one has $U = U_s cup U_{1-s}$.

(ii) For each $U in X$ and each family $(s_i)_{i in I}$ in $A(U)$ generating the unit ideal one has $U = bigcup_{i in I} U_{s_i}$.

Here $ U_s subseteq U$ is the largest subobject on which $s$ is invertible.

But I don’t think that (i) is equivalent to (ii), since (i) is satisfied for $A=0$, right? Notice that (ii) implies that $A(U)=0 implies U=0$ (take $I=emptyset$, cf. MO/45951), which I would expect from a local ring object (see also here).

Am I missing something?