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.


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

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