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.


Question: Why is Nynäshamn a de facto "port" for Stockholm?
Answer: In fact, Nynäshamn certainly is not the "de facto" port of Stockholm, even for international cruise liners.

Quote from the Ports of Stockholm authority web site:


  The Ports of Stockholm comprise a number of port areas, of which
  Värtahamnen, Frihamnen and Loudden, as well as Stadsgården and
  Skeppsbron are the most prominent commercially.  [...]
  
  Värtahamnen, Frihamnen and Stadsgården have extensive ferry services
  for both goods and passengers travelling to and from Finland and the
  Baltic countries. Stadsgården and Skeppsbron are the points of arrival
  and departure for the regular scheduled cruise traffic to and from
  Mariehamn.
  
  In the summer Stadsgården, Skeppsbron and Frihamnen are the ports favoured by the  international cruise liners.


The same site gives a plausible explanation as to why your cruise ship chose to stop at such a remote port (emphasis mine):


  Our most southern port – the Port of Nynäshamn – lies approximately
  sixty kilometres south of Stockholm. Boats transporting both freight
  and passengers depart from Nynäshamn.
  
  [...]
  
  In Nynäshamn we can also receive the larger international cruise
  liners that are not able to, or do not want to, voyage through the
  archipelago into Stockholm. In 2010 for example, a number of vessels
  from the cruise company Norwegian called during May-September.


I think most of them would be able to, because many of the daily ferries to Finland or Baltic countries (such as M/S Viking Cinderella pictured below) are very large vessels too, and they have no problems navigating through the archipelago into the more central ports of Stockholm.

Edit: Well, looking at the map, it's actually quite clear why some briefly visiting cruise ships—arriving from the south—would rather stop at Nynäshamn. They'd have to take a long, time-consuming detour to reach the ports closer to Stockholm (see the maritime routes marked on the map, which approach Stockholm from northeast).


Photo: EnDumEn, Wikimedia Commons


  Or is it an important tourist attraction in its own right that should
  be given equal attention with Stockholm?


Nynäshamn seems to be a nice little seaside town (population around 13,000), but as a tourist attraction it's not on par with Stockholm by any stretch of imagination. However, the Wikipedia article about it claims (no source cited) that it's "a hugely popular destination for Swedes of all ages in the summer months". :-)