How to use universal constructions to create this category/object?

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!

• If you can write down a universal property for what you want, then anything with that property will be unique up to unique isomorphism. The only question is existence. Incidentally, what you are trying to do has nothing to do with delooping. – Zhen Lin Nov 20 '13 at 17:30

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.
• Thanks. That makes a lot of sense. The motivating example for this question is: Consider category of groupoids, if $X$ is a groupoid we can consider it as a groupoid action of some $G$ acting on its objects and we have a monomorphism (faithful functor) from $X$ to $\textbf{B}G$ (tinyurl.com/nswfcnv). I was hoping to generalize this notion to a wider range of categories or at least find a nice way to construct $\textbf{B}G$ from $X$. – Harry Nov 21 '13 at 3:41
• You can also describe $M(C)$ as follows: $B(M(C))$ is the pushout of $1 \leftarrow C^\delta \to C$ where $1$ is the terminal category and $C^\delta$ is the category with the same objects as $C$ but only identity morphisms. – Omar Antolín-Camarena Nov 4 '14 at 17:08