Diagrams in a model category

Question

Let $\mathcal{M}$ be a model category with class of weak equivalences $W$ and $\mathcal{J}$ a small category. The category of diagrams $\operatorname{Fun}(\mathcal{J}, \mathcal{M})$ inherits a class of weak equivalences $W_{\mathcal{J}}$, those natural transformations which are objectwise weak equivalences. If $\mathcal{M}$ fails to be left or right proper then $(\operatorname{Fun}(\mathcal{J}, \mathcal{M}), W_{\mathcal{J}})$ may fail to be a model category. Is it still the case that $\operatorname{Fun}(\mathcal{J}, \mathcal{M})[W_\mathcal{J}^{-1}]$ is locally small? I suspect not but would appreciate a counterexample.