Given a set $X$, we can promote it to a discrete category $\mathcal{C}\_X$ by considering a set containing identities on all the objects of $X$ and trivial composition/identity selecting functions.



> 
> What is a natural way to view this process of adding identities?
> 
> 
> 


Assuming we're working in $ZFC$ (or some other background set theory without atoms) we can take the identiy arrows to be literal identity functions $1\_X$ on the sets $x\in X$ and composition to be literal composition of functions, but this seems ugly from a categorical perspective.


It seems we could also add some primitive objects $1\_x$ to our theory satisfying idempotence under composition, but this feels like we're doing a lot to accomplish something simple.


The motivation for this question is convenience in defining a notion of functor between different dimensions of category. It is common practice in descent theory to consider pseudofunctors from a category $\mathcal{C}$ to a bicategory $\mathfrak{C}$, and these can be defined by promoting ${\bf Hom}\_\mathcal{C}$ to a category as above to promote $\mathcal{C}$ to a $1$-truncated strict bicategory $\mathfrak{C}\_\mathcal{C}$, then considering a pseudofunctor $P:\mathfrak{C}\_\mathcal{C}\to\mathfrak{C}$.


Although I haven't seen it in the literature anywhere (maybe because it isn't interesting), we could also use promotion to define a pseudofunctor from a bicategory $\mathfrak{C}$ to a regular category $\mathcal{C}$ by considering a pseudofunctor $Q:\mathfrak{C}\to\mathfrak{C}\_\mathcal{C}$.


This extends easily to higher dimensions and is all based on this process of 'adjoining identity arrows to a set', so I was hoping for a clear explanation of this process.


It was pointed out in the answer a recent [question](https://mathoverflow.net/questions/377630/discrete-bicategories-n-categories) of mine that trying to 'demote' a category to a set doesn't respect equivalence, but this process of promotion pretty trivially does so hopefully it can be understood easily from a categorical perspective


Any assistance is appreciated.



Categorical promotion
Let me preface this answer by noting that I don't know whether this is true all known models for $n$-categories; certainly it is true in the quasicategorical model for $(n,1)$-categories, and it seems to me like it should be true in all reasonable models and for $(n,n)$-categories as well.


The construction you describe from $n$-categories to $(n+1)$-categories should be right adjoint to the truncation functor from $(n+1)$-categories to $n$-categories - this truncation functor is maybe not so sensible for $(n+1)$-categories that aren't $(n+1,n)$-categories, but it should still be defined - or at the very least, that could give an adjunction between $(n+1,n)$-categories and $n$-categories (and then we could just follow the right adjoint with the inclusion of $(n+1,n)$ into $(n+1)$)


The reason this should be true is of course that if all $n+1$-morphisms in $\mathfrak C\_\mathcal C$ are identities, then any functor $\mathfrak{C\to C}\_\mathcal C$ must factor through the truncation, and uniquely so.


