How to Construct the Coproduct of Pointed Categories? A pointed category is a (small) category together with a distinguished object, called the basepoint $*$. How does one explicitly construct the coproduct of two pointed categories? 
This problem reduces to constructing the coproduct of two pointed connected categories. If both categories have one object, this becomes the free product of monoids.
 A: Well it is the pushout of $C \leftarrow 1 \rightarrow D$. Colimits of categories can be constructed by "merging" generators and relations, as it is the case with every algebraic category. [Of course, here $2$-colimits are pretty natural, but they turn out to be far more complicated!] Explicitly, $C \cup_1 D$ is the following category: It has objects $\mathrm{ob}(C) \cup_{\star} \mathrm{ob}(D)$, where we identify the two base objects $\star$. A morphism from, say,  $c \in C$ to $d \in D$ is a finite chain of morphisms
$c \to c' \to \star \to d \to d' \to \star \to c'' \to c''' \to \star \to \dotsc \to d$.
We have the obvious cancellation rules. When $C,D$ have just one object, namely $\star$, this is the usual construction of the "free product" (this terminology is quite misleading since it is not a product, but rather a coproduct) of monoids.
A: It is useful to consider this construction in more generality, as is well written up in the case of groupoids (see Higgins' book "Categories and groupoids"). Let $C$ be a small category, and $f: Ob(C) \to Y$ a function. Then there is a category $f_*(C)$ with object set $Y$ and satisfying a universal property with regard to functors on $C$ whose object function factors through $f$. If $C,D$ are two categories with base points, then you get the free product by taking the disjoint union of $C$ and $D$ and then identifying the base points. 
To put this in a more general light, the objects of a (small) category can be regarded as a functor Ob from the category $\mathsf{Cat}$ of small categories and functors to the category $\mathsf{Set}$ of sets and functions, and then this functor Ob is both a fibration (this is about pullbacks) and cofibration  (this is about pushouts). There is a basic  account of these ideas, which originated with Grothendieck,  in 
R. Brown and R. Sivera, `Algebraic colimit calculations in
homotopy theory using fibred and cofibred categories',  Theory and
Applications of Categories, 22 (2009) 222-251.
