Colimit

Remarks

Equivalently, a colimit is an initial object of the category of cocones under $D$, so it is unique up to unique isomorphism. A colimit of $D$ in $\mathcal{C}$ is the same as a limit of the corresponding diagram $D^{\mathrm{op}} \colon J^{\mathrm{op}} \to \mathcal{C}^{\mathrm{op}}$ in the opposite category. The colimit of the empty diagram is an initial object.

Example: in the category of sets, every diagram $D$ with $J$ small has a colimit. Let $\sim$ be the smallest equivalence relation on the disjoint union $\bigsqcup_j D(j)$ that identifies $x \in D(j)$ with $D(u)(x) \in D(k)$ for every $u \colon j \to k$. The colimit is the quotient $\left(\bigsqcup_j D(j)\right) / {\sim}$. The injections send $x$ to its class, and a cocone $(N, \psi)$ induces $h([x]) = \psi_j(x)$, which is well defined by the cocone condition.

Non-example: for the diagram of two parallel functions $f, g \colon X \rightrightarrows Y$ with $f \neq g$, the set $Y$ with $\iota_Y = \id_Y$ and $\iota_X = f$ is not a cocone, since $\iota_Y \circ g = g \neq \iota_X$; so it cannot be a colimit.