# Existence of terminal object in cocomplete category

Let $$C$$ be a cocomplete category and $$S$$ be a set such that $$S \subset \operatorname{Ob} C$$ and for any object $$a$$ in $$C$$ there is at least one arow from $$a$$ to some object in $$S$$. Is there terminal object in $$C$$.

I know that answer is yes, but I have some difficulties with understanding why.

Suppose a diagram which consists of all objects in $$S$$ and $$Id$$ arrows. Since this diagram is small there is a colimit $$K$$ with following property: for any object $$a$$ in $$C$$ there is an arrow from $$a$$ to $$K$$. $$K$$ is not a terminal object since there is nothing to guarantee that such arrows are unique.

I was given a hint to take a colimit of diagram with only object $$K$$ and all endomorphisms of $$K$$ as arrows. The claim is that described colimit $$E$$ is terminal object and I can't see why.

Thanks!

• You can find the proof of the dual statement in Mac Lane, "Categories for the Working Mathematician", p.120. – Oskar Feb 11 at 23:15