I suspect that the answer to my question is no, but let me give it a shot anyway.

If $\mathcal{A}$ is a small simplicially enriched category, then the category of simplicial presheaves $\mathsf{sSet}^{\mathcal{A}^{\mathrm{op}}}$ carries two basic model structures: the projective one and the injective one. For each $a \in \mathcal{A}$ the representable presheaf $\mathcal{A}(-, a)$ is both projectively and injectively cofibrant, but usually it is neither projectively nor injectively fibrant.

However, we can replace $\mathcal{A}$ by a "locally fibrant" simplicial category, e.g. $\mathcal{A}' = \mathrm{Ex}^\infty \mathcal{A}$. Then representables over $\mathcal{A}'$ are projectively fibrant.

My question: can we find a simplicial category $\mathcal{A}''$ DK-equivalent to $\mathcal{A}$ such that all representables over $\mathcal{A}''$ are injectively fibrant?


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