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?

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    $\begingroup$ I'm probably missing something, but it seems like that essentially never happens as soon as there exist two non-equal objects which are isomorphic. $\endgroup$ – Dylan Wilson Jul 24 '16 at 14:34
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    $\begingroup$ @DylanWilson Even as soon as there exists a non-identity isomorphism. And vice versa. $\endgroup$ – მამუკა ჯიბლაძე Jul 24 '16 at 15:55

მამუკა ჯიბლაძე's comment is exactly right: this property holds if and only if $C$ has no non-identity isomorphisms (otherwise $(a,\phi)$ for $\phi:a\simeq b$ is an isomorphism in $C\times C$ with codomain in the image of the diagonal but no lift to $C$). But if we only have to worry about identity isomorphisms, then obviously they will always have lifts.

There is a technical term for this definition. When $C$ satisfies this property, it is called a "gaunt" category.

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