TITLE: Can monics always be "made left-invertible" by adding arrows?
QUESTION [12 upvotes]: I am wondering whether there can be categories where monic arrows cannot be "made left-invertible", by embedding the category into a larger category with the left-inverse.
For example, if I have the category with two objects and an arrow between them (and identities), then of course the arrow is monic, and you can just "add in" an arrow going the other direction to invert it. But I am wondering if this is "always possible" in some sense.
More formally,
Let C be a category. Does there necessarily exist a category D which is a "supercategory" of C (i.e. C is a subcategory) such that every monic arrow of C becomes left-invertible in D and every epic arrow becomes right-invertible?
Or is there a counterexample?

REPLY [3 votes]: It may be worth pointing out that the answer to the weaker question of "Is there a faithful functor $F:C\to C'$ sending monos to split monos" is positive. It's usually a better idea to try to only "improve" either limit-type or colimit-type constructions in a category-doing both at once gets you into trouble as in Eric's answer.
Indeed, we can just take the composite of the Yoneda embedding $C\to [C^{\mathrm{op}},\mathrm{Set}]$ with functor of precomposition along the inclusion of the set of objects (viewed as a discrete category) $i:\mathrm{ob}(C)\to C^{\mathrm{op}}$. The resulting functor $i^*:[C^{\mathrm{op}},\mathrm{Set}]\to [\mathrm {ob} C,\mathrm{Set}]$ is faithful, but not full; both the Yoneda embedding and $i^*$ preserve and reflect monomorphisms, and in $[\mathrm {ob} C,\mathrm{Set}]$, which is isomorphism to a product of copies of the category of sets. EDIT: Composing with one more functor the (faithful) disjoint basepoint functor $\mathrm{Set}\to \mathrm{Set}_*$ into the category of pointed sets, gives a faithful functor from $C$ into a product of copies of $\mathrm{Set}_*$, in which every monomorphism splits.