§ Colimits examples with small diagram categories

• Given a colimit, compute the value as taking the union of all objects, and imposing the relation $x \sim f(x)$for all arrows $f \in Hom(X, Y)$ and all $x \in X$.

• A colimit of the form $A \xrightarrow{f} B$ is computed by taking $A \sqcup B$ and then imposing the relation $a \sim f(b)$. This is entirely useless.
• A colimit of the form $A \xrightarrow{f, g} B$ is computed by taking $A \sqcup B$ and then imposing the relation $a \sim f(a)$ as well as $a \sim g(a)$. Thus, this effectively imposes $f(a) \sim g(a)$. If we choose $f = id$, then we get $a \sim g(a)$. So we can create quotients by taking the colimit of an arrow with the identity.
• A colimit of the form $A \xleftarrow{f} B \xrightarrow{g} C$ will construct $A \cup B \cup C$ and impose the relations $b \sim f(b) \in A$ and $b \sim g(b) \in C$. Thus, we take $A, B, C$ and we glue $A$ and $C$ along $B$ via $f, g$. Imagine gluing the upper and lower hemispheres of a sphere by a great circle.