Let $C$ be a category and let $\Delta:C\rightarrow C\times C, \Delta=(id_c,id_c)$ be the diagonal functor.

Recal that an isofibration is a functor p: E→B such that for any object $e\in E $ and any isomorphism $\phi:p(e) \simeq b$, there exists an isomorphism $\psi:e \xrightarrow{\simeq} e'$ such that $p(ψ)=\phi$.

Are there any conditions on $C$ that will ensure the diagonal is an isofibration?