I state what above in your intervention.
And I state what follow:
1] Let $\tau$ the Grothendieck topology on $\mathscr{C}$. Gived a sieve $R\subset X$ (considering it as a full subcategory of $\mathscr{C}\downarrow X$ or a subobjet of $h_X$) we call it a $\tau$-covering (of $X$) if for any sheaf $S$ the restriction morphism $R^\star: S(X)\cong Shv(X, S)\to Shv(R, S) \cong >{\underrightarrow{lim}}_{(y\to X)\in R} S(Y) $ is a isomorphism ($Shv$ mean “sheaves”).
This is equivalent to say one of the following two equivalent condiction:
a) $\iota: R \subset X$ is a Isomorphism in the category $Shv(\mathscr{C}, \tau)$
b) The image of $R \subset_{full} \mathscr{C} \downarrow X $ in $Shv(\mathscr{C} , \tau)$ describes a colimit cocone of $X\in Shv(\mathscr{C} , \tau)$.
The class of all $\tau$-covering define e Grothendieck topology $\widetilde{\tau} $ such taht $Shv(\mathscr{C} , \widetilde{\tau})=Shv(\mathscr{C} , \tau)$, and is the bigger topology with this propriety.
b’) Gived a family $\mathcal{F} =(f_i: X_i\to X)$. The sieve generated is $\tau$-covering iff :
completing $\mathcal{F}$ by all couple af pullback $X_i\times_X X_j\ i,j\in I$ and let $\mathcal{F’}$ the enriched family (observe that the first inclusion $\mathcal{F'}\subset R \subset \mathscr{C} \downarrow X$ is final) then the image of $\mathcal{F’}$ in $Shv(\mathscr{C} , \tau)$ is a colimit cocone (in literature find also a "Pullback invariant condition", in this case this follow automatically) .
Now your request is the following condition:
give a diagram $(X_i \xrightarrow{x_i} C)_{i\in I}$
(dont write transitions morhisms) and let $X:= {\underrightarrow{lim}}_{I} X_i$ in $\mathscr{C}$, the natural morphism
$y(X) \to {\underrightarrow{lim}}_{I} y(X_i) $ is a isomorphism in $Shv(\mathscr{C} , \tau)$.
Infact we state that:
2] Considering that in any category $\mathscr{C}$ the proiection funtor $\pi : \mathscr{C}\downarrow X \to \mathscr{C} $ create colimits (i.e. make o colimit in the comme $\mathscr{C}\downarrow X$ is “the some” that make the some colimit in $\mathscr{C}$).
Then if in your data the object $C$ isnt fixed but generic your request is equivalent to the follow:
give a diagram $(X_i)_i$
and let $X:= {\underrightarrow{lim}}_{I} X_i$ the natural morphism
$y(X) \to {\underrightarrow{lim}}_{I} y(X_i) $ is a isomorphism in $Shv(\mathscr{C} , \tau)$.
(you can put $C:= X$).
Then form 1-(b’) above this is neccessary that the colimit cocone $X_i \to X$ generate a $\tau$-covering, then we can state the condiction as follow:
give a colimit cocone $(X_i \to X)_{i\in I}$
and suppose that it generate a $\tau$-sieve,
then ${\underrightarrow{lim}}_{I} y(X_i) \to y(X)$ is a isomorphism in $Shv(\mathscr{C} , \tau)$?
But this is equivalent to the condiction $\tau$ is sub-canonical, i.e. any representable presheaf $h_X$ is a sheaf or equivalently any cover is a colimit (or precover (completated by pullbak’s) if we start from a pretopology).
Of course this happen for topological covering (any open topological covering is a colimit too)
