TITLE: If $\mathcal{K}$ is well behaved and has connected limits then $\mathcal{K}_{/K}$ has pushouts?
QUESTION [1 upvotes]: Is the following statement true?

Let $\mathcal{K}$ be a category such that:

$\mathcal{K}$ is well powered;

every map is a monomorphism;

$\mathcal{K}$ has connected limits.


Then for any object K the category $\mathcal{K}_{/K}$ has pushouts.

I also worked out a candidate proof for it, but it really looks to me like a strange result.
Proof:
The idea of this proof is that a pushout is the infimum of a family.
Consider a diagram $A \leftarrow  B  \rightarrow C$ of objects in $\mathcal{K}_{/K}$. The class of cocones $\{L_i\}$ is a family of subobjects of $K$. Call $L^{\infty}$ the wide pullback of this family, this object exists because the category is wellpowered. $L^{\infty}$ is the desired pushout.

REPLY [2 votes]: Yes, this is correct, and in fact the same argument shows $\mathcal{K}_{/K}$ has all colimits.  Something that may make this easier to swallow is to observe that your hypotheses imply $\mathcal{K}_{/K}$ is a preorder (and essentially small).  Indeed, if $a:A\to K$ and $b:B\to K$ are two objects and $f,g:A\to B$ are morphisms of $\mathcal{K}_{/K}$ (so $bf=bg=a$), then $f=g$ since $b$ is a monomorphism.  Moreover, every limit in $\mathcal{K}_{/K}$ can be represented as a connected limit in $\mathcal{K}$, so $\mathcal{K}_{/K}$ has all limits.  Your result is then just a special case of the fact that a (small) poset which is meet-complete is automatically also join-complete.
(In fact, the same argument would show that any complete essentially small category is automatically cocomplete.  This is not actually any more general, though, since any complete essentially small category must be a preorder.)
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