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?



Isofibrations and Diagonal Functors
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.


