Let $C$ be a category with two different topologies $\tau_1, \tau_2$ such that $\tau_1$ is stronger, i.e. any covering in $\tau_2$ is also a covering in $\tau_1$. Denote the corresponding sites by $C_1, C_2.$ Let $F \in Sh(C_1)$, in particular, $F$ is also a sheaf with respect to $\tau_2$. Definitely, we should not have $H^i (C_1, F) = H^i (C_2, F)$ but I am confused because:
Choose an injective resolution $F \to I_0 \to I_1 \to .. $ where $I_j \in Sh(C_1)$ are injective sheaves.
Each $I_j$ is also in $Sh(C_2)$ and is injective.
Suppose $C$ has a final object $X$, then we just compute cohomology of $I_0 (X) \to I_1 (X) \to ..$ and by construction they compute both $H^i (C_1, F)$ and $H^i (C_2, F)$.
Where is the mistake?