Im supposed, in the category $\textbf{Mon}$ of monoids, prove that all equalisers and finite products exist. Does this mean that I have to prove that every pair of arrows has an equaliser, that the category has a terminal object and that the category has binary products?
In my book, I read that a terminal object is a product of no objects, can someone explain this? IS this why we require a terminal object in proving that a category has ALL finite products?
Define the product of the set $\{a_1,a_2\dots,a_n\}$ as $a_1a_2\dots a_n$.
What should we mean by the empty product, i.e. the product of $\emptyset$?
If it will have a value, we have to define it as $\bf1$ to keep the properties, as $1$ doesn't affect the product: $1\cdot x=x$ for any number $x$.
The same goes on for the categorial product, of which the terminal object $\bf1$ is the unit, in that ${\bf1}\times X\cong X$ for any object $X$.
Having all finite products means that the product exists for all finite sets of objects.
$\emptyset$ is a finite set of objects.