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 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.