Given an a category $\mathcal{C}$ what is the most concise way to construct the smallest one object category $\mathcal{D}$ such that there exists a faithful functor from $\mathcal{C} \to \mathcal{D}$? Is there some nice universal construction which gives me a unique $\mathcal{D}$ and faithful $F:\mathcal{C} \to \mathcal{D}$?
Is there an analogue for this construction in a category (perhaps being a 2-category or monoidal is needed)? One object category would be replaced with an object with one morphism from the terminal object and faithful functor with monomorphism?
I'm very new to category theory is this related to the delooping of an object $X$?
Many thanks!
Let $\mathcal{C}$ be a small category. Let $M(\mathcal{C})$ be the free monoid generated by symbols $[f]$ for every morphism $f$, modulo the relations: $[\mathrm{id}]=1$ and $[f \circ g]=[f] [g]$ for composable morphisms $f,g$. Then $M : \mathsf{Cat} \to \mathsf{Mon}$ is left adjoint to the delooping functor $B : \mathsf{Mon} \to \mathsf{Cat}$.
Now the question is if the unit of the adjunction $\mathcal{C} \to B(M(\mathcal{C}))$ is a faithful functor, i.e. if $f,g$ are parallel morphisms in $\mathcal{C}$ such that $[f]=[g]$ in $M(\mathcal{C})$, then $f=g$ in $\mathcal{C}$. Probably one can show this using the (inductive) definition of equality in $M(\mathcal{C})$. Roughly, $M(\mathcal{C})$ only forgets source and target maps of the category, the rest is preserved.