I have a question, I applied for the Schengen visa through Germany, on the form they ask me when I'd arriving (to Germany) and when I'd returning. they take this exact date as the visa period.

I wondering if they would let me get into the plane, even though my flight it a day before the arrival.

Let get the real deal; let say this:

My visa is valid from 02/12/2014 to 02/02/2015. But, my flight is scheduled to be departing on 01/12/2014, but entering Germany 02/12/2014. the flight duration is almost 9hours and there is 7hours difference between Germany and the departing flight country.


Question: How to Construct the Coproduct of Pointed Categories?
Answer: 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.