A monomorphism from a colimit Let $D$ be an upward directed poset, and suppose I have a diagram $F: D\to C$ in a cocomplete category such that $F(d)\to c$ is a monomorphism for all $d\in D$ for some $c\in C.$ Is it true that $\operatorname{colim}_DF\to c$ is also a monomorphism?
When $D$ is the empty category this is vacuously true. But in general, I don't how this works. An insight to answer this question would be greatly appreciated.
 A: It is not true that this is vacuously true when $D=\emptyset$ !
In fact there are counterexamples. Indeed, if $D=\emptyset$, $colim_D F$ is an initial object of $C$, so you're asking : is $\emptyset \to c$ always a monomorphism ?
The answer is no: in the category of (unital) rings, $\emptyset = \mathbb Z$ and you have a morphism $\mathbb Z\to \mathbb Z/p$ which is not a monomorphism (left as an exercise)
If you allow $D$ to be nonempty, there will be more examples. For instance, let's dualize, and ask : if every $c\to F(d)$ is an epimorphism, is $c\to\lim_{D^{op}} F$ an epimorphism ?
The answer is more intuitively clear: no in general. For instance, let $D= \mathbb N$ and the functor $D^{op}\to Ab$ sends $n\mapsto \mathbb Z/p^n$, with the canonical projection maps for $n\leq m$. Then $\mathbb Z\to \mathbb Z/p^n$ is an epimorphism for all $n$, but $\mathbb{Z\to Z}_p := \lim_n \mathbb Z/p^n$ is not.
Suppose however that you have a "forgetful" functor $U: C\to Set$ which preserves nonempty directed colimits, and such that $f$ is a monomorphims if and only if $U(f)$ is. Then the claim holds in $C$ (because it does in $Set$) - so most algebraic categories work in this respect (note that in the previous example, we had $C=Ab^{op}$, and not $Ab$)
