Proving $ \{ \langle D_1, … ,D_K \rangle : \text{ where } D_i \text{ are DFAs and } {\bigcap}_{i=1}^k L(D_i) = \emptyset \} $ is NP-Hard

Question

The question (Prove L is NP-hard) was about proving that the following language is NP-hard: $$ L = \{ \langle D_1, D_2, ... ,D_K \rangle : k \in {N}\text{, the } D_i \text{ are DFAs and } {\bigcap}_{i=1}^k L(D_i) = \emptyset \} $$

That got me thinking about the related problem:

$$ L' = \{ \langle D_1, D_2, ... ,D_K \rangle : k \in {N}\text{, the } D_i \text{ are DFAs and } {\bigcap}_{i=1}^k L(D_i) \neq \emptyset \} $$

I would imagine that $L'$ is also NP-hard, but I couldn't think of any reductions.. am I missing something obvious?